\appendix
\chapter[Green's function for the Laplacian in $2d$]{Green's function for the
Laplacian in two dimensions \label{app:Green}}
\mychapter
In this appendix we provide some useful formulas that can be derived from the
Green's function of the Laplacian\footnote{To get the normalisation constants
right, I checked most of the formulas
for some test functions by using numeric integration.}. We will first show
that in two dimensions this Green's function is given by
${1\over4}\log|x-x_0|^2$. We need to prove\col
\be
{1\over4}\nabla_0^2 \int dx^2\, f(x) \log |x-x_0|^2 = \pi f(x_0)\,
.\label{eq:Greenx}
\ee
For this we write the integral in cylinder coordinates around $x_0$. We get
for the \lhs\col
\[
{1\over4}\nabla_0^2 \int dr d\theta\,f(x+x_0) 2r \log r \, .
\]
We can bring the Laplacian inside the integral, and let it act on
$x$\col
\[
{1\over2}\int dr d\theta\, r \log r \left(
  {1\over r} {\partial \over \partial r} r{\partial f(x+x_0)\over \partial r} +
  {1\over r^2}{\partial^2 f (x+x_0)\over \partial \theta^2}
  )
\right)\, .
\]
The second part of the integral is zero because of the total derivative in
$\theta$. The first part can easily be evaluated to\col
\[
{1\over2}\int dr d\theta\, {\partial \over \partial r}\left(
    r \log r {\partial f(x+x_0)\over \partial r} - f(x+x_0)\right)\, .
\]
Assuming that $f$ decays fast enough to zero at infinity (we call the set of these functions
$\cs$), we get the desired result \re{eq:Greenx}.

Written in terms of the $z,\zb$ coordinates, \vgl{eq:Greenx}
becomes\col \be
{\partial \over \partial \zb_0}{\partial \over \partial z_0}\int dx^2\, f(z,\zb)
\log (z-z_0)(\zb-\zb_0) = \pi f(z_0,\zb_0)\label{eq:Greenz}\, ,
\ee
or also
\be
\int d^2x\, {\partial f \over \partial \zb}(z,\zb)\,(z-z_0)^{-(n+1)} = -\pi
{1\over n!}{\partial^n f\over \partial z_0^n}(z_0,\zb_0)\,,\,\,\forall n\in \Bbb{N}
\label{eq:GreenzGen} \ee
where we have generalised the result by taking additional derivatives with
respect to $z_0$ and changed the derivatives with respect to $z_0,\,\zb_0$ to
$z,\,\zb$. This is often written as an equality between
distributions\col
\be
{\partial \over \partial \zb_0}{1\over (z-z_0)^{n+1}} = -\pi
{1\over n!}{\partial^n \over \partial z_0^n}\delta^{(2)}(x-x_0)\label{eq:distributionzz0}\, ,
\ee
where the integration over $z,\,\zb$ is implied.

If $f$ is analytic in an neighbourhood of the point $(z_0,\zb_0)$ (it cannot
be analytic in the whole complex plane as it has to be an element of
$\cs$),  we can use the Cauchy residue formula to rewrite
\vgl{eq:GreenzGen}\col
\be
{1\over\pi}\int d^2x\,
   {\partial f \over \partial \zb}(z,\zb)\,(z-z_0)^{-(n+1)} =
- \oint_{C_0} {dz\over2\pi i}\,
   f(z)\,(z-z_0)^{-(n+1)}\, ,\label{eq:dbarToContour}
\ee
where the contour $C_0$ surrounds $z_0$ anti-clockwise and lies in the region
where $f$ is analytic. This important formula is used in the subsection on
Ward-identities \ref{ssct:WardId}.

Eq.\ \re{eq:GreenzGen} can be used to define an inverse derivative.
\puzzle{$f\in \cs\rightarrow g\in\cs$?}
Indeed, for $n=0$ we see that (using the notation $\db \equiv {d \over
d\zb}$)\col
\be
\int d^2x\, f(z,\zb)\,(z-z_0)^{-1} = -\pi g(z_0,\zb_0) \Rightarrow \db g=f  \, ,
\ee
for $f$ such that (\ref{eq:distributionzz0}) is valid for $\db f$, or $\db f
\in {\cal S}$. We define\col
\be
\db^{-1} f(z_0,\zb_0) = - {1\over\pi} \int d^2x\, f(z,\zb)\,(z-z_0)^{-1}
\, .\label{eq:inversedb}
\ee


\chapter{Superconventions \label{app:super}}
\noindent{\sc Indices}\\[2mm]
\mychapter
We use the conventions of \cite{deWitt:indices} when working with
supervectors, -matrices, and the like. Left and right indices occur,
an element of a supermatrix is for instance ${}_iA^j$. Summation is
implied over indices which occurs twice, where no signs are introduced
when indices are near each other and one is on top, the other down, \eg
$b^i\ {}_iA^j$. We use the two following rules for shifting indices\col
\bea
A^i = (-1)^{Ai}\ {}^iA&&A_i=(-1)^{(A+1)i}\ {}_iA\,,\label{eq:indexShift}
\eea
where in exponents of $-1$, $A$ denotes the parity of the ``symbol"
$A$ and $i$ the parity of the index, such that an element of a
supermatrix  ${}_iA^j$ has parity $(-1)^{A+i+j}$. For instance one
could have a bosonic matrix where the indices take values in a bosonic
and fermionic sector. The rules \re{eq:indexShift} are such that\col
\be A^i\ {}_iB = (-1)^{AB} B^i\ {}_iA\,.\label{eq:superSwap}
\ee
\vspace{3ex}
\noindent{\sc (Super)Lie algebras}\\[2mm]
We use in chapter \ref{chp:WZNW} and \ref{chp:renormalisations}
realisations of a super Lie algebra $\bar{g}$. The generators (which are
matrices with numbers as elements) are denoted by ${}_at$, $a\in\{ 1,
\cdots, d_B+d_F\}$, where $d_B$ ($d_F$) is the number of bosonic
(fermionic) generators. We take $t$ a bosonic symbol. We will always
assume that the representation is such that ${}_a t$ is a matrix
containing numbers (zero parity). This means that ${}_i(_at)^j$ is zero
unless $(-1)^{a+i+j} = 1$. We denote the (anti)commutation relations
by\col
\be
[{}_a t, {}_b t]={}_a t\ {}_b t-(-)^{ab}{}_b t\ {}_a t={}_{ab}f^c
\ {}_c t\,, \ee
where ordinary matrix multiplication is used. The structure constants
are thus also numbers, and ${}_{ab}f^c$ is zero except if 
$(-1)^{a+b+c}=1$.

When we have a super Lie algebra valued field $A$, its components
are\col
\be
{}_iA^j = {}_i(A^a\ {}_at)^j \equiv (-1)^{i(A+a+1)} A^a\ \,{}_i(_at)^j\,,
\label{eq:matrixconv}
\ee
where the last equation follows the convention \re{eq:indexShift}.
This gives us for the elements of the matrix (anti)commutator\col
\be
{}_i([A,B])^j = {}_i\left(-(-1)^{aB} A^aB^b  \ {}_{ba} f^c\ {}_ct\right)^j\,.
\ee

From the Jacobi identities, one shows that the adjoint representation is given
by\col
\be
{}_b(_at)^c\equiv {}_{ba}f^c\,.
\ee
The Killing metric ${}_a g_b$ is defined by
\be
(-1)^d\ {}_{ca}f^d\ {}_{db}f^c = - \tilde{h}\ {}_ag_b\,,
\label{eq:Killingdef}
\ee
where $\tilde{h}$ is the dual Coxeter number. Though this is perfect for
ordinary Lie algebras, this is not sufficient for super algebras as the
dual Coxeter number might vanish in this case\footnote{The possibility of
a vanishing quadratic Casimir in the adjoint representation, has the
interesting consequence that in that case,  the affine Sugawara
construction always gives a value for $c$ which is independent of the
level of the affine super Lie algebra. E.g.\ for $D(2,1,\alpha)$, one always
has $c=1$.}. More generally, it is defined via the supertrace\col
\be
\str(_at\ {}_bt)\equiv
{}_i(_at)^j \ {}_j(_bt)^i (-1)^{i} \equiv -x\ {}_{ab}g\,,
\label{eq:strdef}
\ee
where $x$ is the index of the representation. Obviously we have
$x=\tilde{h}$ in the adjoint representation. Note that using
\vgl{eq:matrixconv}, the supertrace of a product of two Lie algebra valued
fields is\col
\be
\str( A B) \equiv {}_i (AB)^i (-)^{i(A+B+1)} =
 -x A^a\ {}_ag_b\ {}^bB = (-1)^{AB} \str( B A)\,.
\ee
The Killing metric is used to raise and lower indices\col
\be
A^a=A_b\ {}^bg^a,\qquad A_a=A^b\ {}_bg_a\,,
\ee
where ${}^ag^b$ is the inverse of ${}_ag_b$\col\ ${}^ag^c\
{}_cg_b={}^a\delta_b$.

Table \ref{table:LieAlg} contains some properties of the (super) Lie
algebras which appear in chapter \ref{chp:renormalisations}. We denote by
$x_{\rm fun}$, the index of the fundamental (defining) representation.
For $D(2,1,\alpha)$, it is not clear how to define the fundamental
representation. The size of the smallest representation of $D(2,1,
\alpha)$ depends on $\alpha$. The smallest representation which exists
for generic values of $\alpha$ is the adjoint representation.
\begin{table}[hbt]
\caption{Properties of some (super) Lie algebras.\label{table:LieAlg}}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|}  \hline
 & & & & & \\
algebra  & bosonic & $d_B$    & $d_F$ & $\tilde{h}$ &$x_{\rm fun}$\\
 & subalgebra& && &\\*[1mm] \hline
{$sl(n)$}  & {$sl(n)$} &{${\scriptstyle n^2-1}$}  &{${\scriptstyle 0}$}&
  {${\scriptstyle n} $}&
  {${\scriptstyle \frac 1 2} $}  \\*[1mm] \hline
{$so(n)$}  & {$so(n)$}  &{$ {\scriptstyle \frac 1 2 n(n-1)}$} &
  {$ {\scriptstyle 0}$} & {${\scriptstyle n-2 }$}   &
  {$ {\scriptstyle 1}$}          \\*[1mm] \hline
{$osp(n|2)$}&{$sl(2)+so(n)$}& {$ {\scriptstyle \frac 1 2 (n^2-n+6) }$}&
  {${\scriptstyle 2n }$} & {${\scriptstyle \frac 1 2 (4 - n) }$} &
  {$ {\scriptstyle \frac 1 2 } $}\\*[1mm] \hline
{$D(2,1,\alpha)$}&{$sl(2)+su(2)+su(2)$}&{${\scriptstyle 9}$} &
  {${\scriptstyle 8} $} & {$ {\scriptstyle 0}$} &
  {${\scriptstyle -}$}           \\*[1mm] \hline
{$su(1,1|2)$}&{$sl(2)+su(2)$}&{${\scriptstyle 6}$} &
  {${\scriptstyle 8} $} & {$ {\scriptstyle 0}$} &
  {${\scriptstyle \frac 1 2 }$}  \\*[1mm] \hline
{$su(m|n)$}&{$su(m)+su(n)+u(1)$}&
  {${\scriptstyle m^2+n^2-1}$} &{$ {\scriptstyle 2mn} $} &
  {${\scriptstyle m-n}$} & {${\scriptstyle \frac 1 2 }$} \\ {$m\neq n$}&
   & & & & \\  \hline
\end{tabular}
\end{center}
\end{table}

\vspace{3ex}
\noindent{\sc $\slt$ embeddings of (super)Lie algebras}\\[2mm]
We now fix our conventions for embeddings of $\slt$ in a (super) Lie
algebra $\bar{g}$. We denote the $\slt$ generators with $\{e_0,e_\pm\}$.
We use normalisations for the $\slt$ generators such that\col
\bea
[e_0,e_\pm ]=\pm 2 e_\pm\,,&& [e_+,e_-]=e_0\,,
\label{eq:RNsl2def}
\eea
with supertraces \re{eq:strdef}\col
\be
\str(e_0e_0)=2\str(e_+e_-)=4xy\,, \label{eq:RNindexydef}
\ee
with $y$ the index of the $\slt$ embedding.

We define the kernels\col
\be
\KER{\pm}\equiv \mbox{ker\,ad\,} e_\pm\,.
\ee


Any highest weight representation $L (\Lambda )$ of $\bar{g}$ decomposes
according to irreducible $\slt$ representations \cite{dynkin}\col
\bea
L (\Lambda ) = \bigoplus_{j\in \frac 1 2
       {\bf N}}n_j(\Lambda )\ \underline{2j\!+\!1}\,,\label{eq:RNBbranch}
\eea
where $n_j(\Lambda)$ denotes the multiplicity of the  $\slt$
representation. Taking \vgl{eq:RNBbranch} for the adjoint
representation of $\bar{g}$,
we can make a choice for the generators of $\bar{g}$ which reflects
the $\slt$ embedding\col\ $\{t_{(jm, \alpha_j)}\}$ where $j\in\frac 1
2{\bf Z}$, $m=-j, -j+1,\cdots , j$ and $\alpha_j = 1, \cdots
n_j(\mbox{\small adjoint})$. We take for the $\slt$ generators
$e_m=t_{(1m,0)}$.\footnote{This differs with a factor $\sqrt{2}$ from
\cite{us1} for $m=\pm1$.}

The algebra $\bar{g}$ acquires a grading\col
\bea
\bar{g}=\bigoplus_{m\in\frac 1 2 {\bf Z}}\bar{g}_m\quad\mbox{
where }\quad \bar{g}_m =\{ a\in \bar{g}\, |\, [e_0,a ] = 2m a\}\,.
\eea
We use the symbol $\P$ for projection operators in $\bar{g}$. $\PHW$
denotes a projection on $\KER{+}$, and $\PLW$ on $\KER{-}$. We also use
$\P_{\geq n}\bar{g}=\oplus_{m\geq n}\bar{g}_m$, $\P_n\bar{g}=\bar{g}_n$,
etc. Furthermore, $\P_+\equiv\P_{>0}$ and $\P_-\equiv\P_{<0}$.

\chapter{A \Mathematica\ primer \label{app:Mathematica}}
\setcounter{mathline}{0}
\mychapter
This appendix gives a short introduction to \Mathematica.
We refer to \cite{Wolfram} for further details.

Symbols in \Mathematica\ are case sensitive. Expressions, except for some
built-in types like {\tt Integer} and {\tt Symbol}, have always the same
structure\col\ they consist of a head and some (possibly null) arguments.
The head and arguments are again expressions. Application of
a head on some arguments is denoted with square brackets. As an example\col
\\ \inm{FullForm[1+a+b]}
\\ \outm{Plus[1,a,b]}\\
One extracts the $n$th subexpression using {\tt Part[expr,n]}, or {\tt
expr[[n]]}.

One can attach transformation rules to any symbol which tell how to
transform an expression with the symbol as head or as one of its
arguments. Evaluation of an expression proceeds by evaluating
first the head, then the arguments (unless some attributes are assigned
to the head). After this, \Mathematica\ checks if any transformation
rules for the expression can be found. It first checks rules assigned to
the arguments ({\tt UpValues}), and then rules assigned to the head
({\tt DownValues}).

Programming in \Mathematica\ is effectively done by specifying a set of
transformation rules. When the \lhs\ of a rule matches the expression
that is being evaluated (taking into account any conditions specified
with {\tt /;}), the transformation rule is applied and the result is
again evaluated. Evaluation continues until no further rules
apply. The most specific rules are used
first\footnote{When \Mathematica\ is not able to figure out an order
between two rules, it checks the rules in the order they were
defined.}, \ie if a rule is given that any expression with head {\tt
f} is zero, and another rule specifies that {\tt f[0]} is $1$, then
the latter rule will be checked first.

Before giving an example it is necessary to discuss the pattern matching
which is used in \Mathematica. The purpose of a pattern is to specify the
conditions when a certain transformation rule has to be used.
We give a list of some frequently occuring patterns.
In this table {\sl p1, p2} stand for any pattern.
%Table \ref{table:patterns} gives a list of some frequently occuring patterns.
%\begin{table}[hbt]
%\caption{\Mathematica\ patterns. In this
%table {\sl p1, p2} stand for any pattern. \label{table:patterns} }
%\vspace{0 cm}
\begin{center}
\begin{tabular}{|c|c|l|}
\hline
                      &             &   \\*[-.3cm]
pattern &    explanation     &  example \\
                      &             &   \\*[-.3cm]
\hline
                      &             &   \\*[-.3cm]
\verb|_|   & any expression  &  {\tt f[1][1,2,3]}\\
\verb|_f|  & any expression with head {\tt f} & {\tt f[1,2,3]} \\
\verb|__|  & sequence of expressions (length $\geq 1$)&{\tt 1,2,3}\\
\verb|___| & sequence of expressions (length $\geq 0$)&{\tt 1,2,3}\\
\tt f[{\sl p1},{\sl p2}] &
         expression with head {\tt f} whose &{\tt f[x,1]} matches\\
           &\ \ \ arguments match {\sl p1,p2} &
                            \ \ \ \verb|f[_,_Integer]|\\
\hline
\end{tabular}
\end{center}
%\end{table}
One can give a pattern a name by prepending it with the name and a
colon, \eg \verb|a:f[_,_]|. An abbreviation for this syntax is possible
when the pattern begins with an underscore, \eg \verb|a_Integer|. Named
patterns are useful first of all to name arguments of a transformation
rule (see below). Additionally, when a named pattern occurs more than
once, all matching items have to be identical, \eg
\verb|f[a_,a_]| matches {\tt f[1,1]}, but not {\tt f[1,2]}.

Let us present a small example to show how all these things fit together
to make a very powerful programming language. The factorial function
could be defined as follows\col
\\ \inm{factorial[n\un Integer] := n factorial[n-1]}
\\ \inm{factorial[0] = 1;}\\
As explained above, the order in which these statements are given is not
important. It is quite simple to make that factorials of an expression
plus a small integer should be transformed into a product\col
\\ \inm{factorial[n\un  + m\un Integer] :=}
\\ \contm{\ \ \ \ factorial[n] Product[n+i,\{i,1,m\}] /; 0<=m<=10}\\
Here the notation {\tt /;} is used to specify a condition. These rules were all
concerning transformations of expressions with head {\tt factorial}.
However, we can also
attach a rule to {\tt factorial} to handle quotients\col
\\ \inm{factorial /: factorial[n\un ] / factorial[m\un ] :=}
\\ \contm{\ \ \ \ Pochhammer[m+1,n-m]}\\
where {\tt Pochhammer} is an internal function corresponding to the
Pochhammer symbol defined in appendix \ref{app:Combinatorics}.

Note that \Mathematica\ does not enforce the use of types
like {\sl Axiom}, but patterns can be used to simulate this.

It is sometimes useful to have a set of transformation rules which is
not applied automatically. Such local rules are normal \Mathematica\
expressions {\tt Rule[{\sl pat},{\sl expr}]}, with alternative
notation {\sl pat} {\tt ->} {\sl expr}. They are used as follows\col
\\ \inm{x + y /. x -> z}
\\ \outm{y + z}\\
Two different assignments are possible in \Mathematica. With {\tt Set}
(or {\tt =}) the \rhs\ is evaluated when the assignment is done (useful
for assignment of results), while with {\tt SetDelayed} (or {\tt :=}) the
\rhs\ is evaluated when the transformation rule is used (useful for
function definitions). A similar difference exists between {\tt Rule}
({\tt ->}) and {\tt RuleDelayed} ({\tt :>}).

We also need some {\sl lisp} and {\sl APL}-like functions
which are heavily used in \OPEdefs.
%They are listed in table \ref{table:Map&others}.
%\begin{table}[hbt]
%\caption{A few functional programming
%functions.\label{table:Map&others} } \vspace{0 cm}
\begin{center}
\begin{tabular}{|l|c|l|l|}
\hline
                      &             &   &\\*[-.3cm]
function &    abbreviation     &  example in& example out \\
                      &             &   &\\*[-.3cm]
\hline
                      &             &   &\\*[-.3cm]
\tt Map   & \tt /@  &  \tt f /@ \{1,2,3\} & \{f[1],f[2],f[3]\}\\
\tt Apply   & \tt @  &  \tt f @ g[1,2,3] & f[1,2,3]\\
\hline
\end{tabular}
\end{center}
%\end{table}
{\tt Scan} is like {\tt Map} but has only side-effects, \ie the
function is applied on the elements of the list, but the results are
discarded.

Finally, when defining a more complicated transformation rule, we will
need local variables. This is done with the {\tt Block} statement which
has the syntax {\tt Block[\{{\sl vars}\}, {\sl statement}]}, where {\sl
vars} is a list of local variables (possibly with assignments) and {\sl
statement} can be a compound statement, \ie statements separated with
a semicolon. The value of the block is result of the statement. Hence a
function definition could be\col
\\ \inm{f[x\un] := Block[{a = g[x]}, a + a\ha 2]}
\\
Note that \Mathematica\ 2.0 introduced a similar statement {\tt
Module}. It makes sure that there is no overlap between globally
defined symbols and the local variables. However, this introduces
considerable run-time overhead compared to {\tt Block}. When a function
is defined in the {\tt Private`} section of a \Mathematica\ package, no
conflict is possible, and {\tt Block} is to be preferred.

\section*{An example : generating tube plots}
We will first define a general purpose function {\tt
TubePlot} which generates a three dimensional plot of a tube, which is
specified by a parametric curve in three dimensions, like in {\tt
ParametricPlot3D}. Then, we will use this function to generate a surface
representing a second order Feynman ``diagram'' for the interaction of two
closed strings, which can be found at the start of this book.

The algorithm for {\tt TubePlot} is to construct at some points a circle
perpendicular to the tangent vector of the curve ({\tt TubeCircle}).
These circles are then sampled and the samples are returned by {\tt
TubePlot}. The function {\tt ListSurfaceGraphics3D} can then be used to
visualise the plot.

{\footnotesize
\begin{verbatim}
Needs["Graphics`Graphics3D`"]
rotMatrix[{a_,b_,c_}] :=
   Block[{sqrab=Sqrt[a^2+b^2],sqrabc=Sqrt[a^2+b^2+c^2]},
     If[N[sqrab / sqrabc] < 10^-6,
       {{-1,1,0}/Sqrt[2],{-1,-1,0}/Sqrt[2],{0,0,1}},
       {{-b,a,0}/sqrab,
        {-a c,-b c,a^2+b^2}/sqrab/sqrabc,
        {a,b,c}/sqrabc}//Transpose
     ]
   ]
TubeCircle[r0_,rp0_,R_] :=
   Evaluate[rotMatrix[rp0] . {R Cos[#], R Sin[#],0} + r0]&
Options[TubePlot] = {PlotPoints->15, Radius->.5,PointsOnCircle->5};
TubePlot[f_,{t_,start_,end_},opts___Rule] :=
   TubePlot[f,D[f,t], {t,start,end},opts]
TubePlot[f_,fp_,{t_,start_,end_},opts___Rule] :=
   Block[{ nrtpoints,nrcpoints, R, tcircle, circlepoints, tval},
     {nrtpoints, nrcpoints, R} =
         {PlotPoints, PointsOnCircle, Radius}/.{opts}/.Options[TubePlot];
     circlepoints = N[Range[0,nrcpoints]/nrcpoints 2Pi];
     Table[
        tcircle = N[TubeCircle[f/.t->tval, fp/.t->tval,R]];
        tcircle /@ circlepoints,
        {tval,N[start], end, N[(end-start)/nrtpoints]}
     ]
   ]
\end{verbatim}
}

We wish to use {\tt TubePlot} with a smooth curve through some points.
The curve can be constructed using cubic spline interpolation, which is
defined in the {\tt NumericalMath`SplineFit`} package. Unfortunately,
this package does not define the derivative of the fitted spline. We can
do this ourselves fairly easily because \verb&SplineFunction[Cubic,__]&
simply stores the coefficients of the cubic polynomials used in the
interpolation.

{\footnotesize \begin{verbatim}
Needs["NumericalMath`SplineFit`"];
Derivative[1][SplineFunction[Cubic,se_,pts_,internal_]]:=
   Block[{int = Map[{#[[2]], 2 #[[3]], 3 #[[4]], 0}&, internal, {2}]},
      SplineFunction[Cubic,se,
            Append[ {#[[1,1]],#[[2,1]]}& /@ int,
                    Apply[Plus,int[[-1]],1]
            ],
            int]
   ]
\end{verbatim} }
Finally, we need a suitable list of points. A last catch is that
{\tt SplineFit} returns a parametric curve in the plane, while {\tt
TubePlot} expects a curve in 3D.
{\footnotesize \begin{verbatim}
pts={{3,0},{2,0},{2,1},{2.6,1},{2,1},{2,2},{1,2},{1,1},{.8,1.8},{0,2},
     {0,0},{.7,0},{1,1},{1,-.2},{1.1,1},{1.1,1.1},{2,1},{2,2},{3,2}};
sf=SplineFit[pts,Cubic];             dsf=sf';
f1[t_?NumberQ]:= Append[sf[t], 1]    f2[t_?NumberQ]:= Append[dsf[t], 0]
tplt=TubePlot[f1[t],f2[t], {t,0,Length[pts]-1.1},
      PointsOnCircle->13,Radius->.25,PlotPoints->70];
ListSurfacePlot3D[tplt,ViewPoint->{0,0,2},
  AmbientLight->GrayLevel[.1],RenderAll->False,Boxed->False];
\end{verbatim}
}
\noindent{\bf Warning:} The CPU time to generate this plot is fairly
small. However, rendering the plot can take several hours depending on
your platform.



\def\AP#1{Ann.\ Phys.\ {\bf #1}}
\def\CMP#1{Comm.\ Math.\ Phys.\ {\bf #1}}
\def\CQG#1{Class.\ Quantum Grav.\ {\bf #1}}
\def\IMP#1{Int.\ J.\ Mod.\ Phys.\ {\bf #1}}
\def\JMP#1{J.\ Math.\ Phys.\ {\bf #1}}
\def\MPL#1{Mod.\ Phys.\ Lett.\ {\bf #1}}
\def\NP#1{Nucl.\ Phys.\ {\bf #1}}
\def\PR#1{Phys.\ Rev.\ {\bf #1}}
\def\PRp#1{Phys.\ Rep.\ {\bf #1}}
\def\PL#1{Phys.\ Lett.\ {\bf #1}}

\begin{thebibliography}{999}
\itemsep -1ex plus 0pt minus 0pt
\frenchspacing
\addcontentsline{toc}{chapter}{Bibliography}

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   M. Ademollo et al., {\sl Supersymmetric strings and colour
   confinement}, \PL{B62} (1976) 105.\\
   M. Ademollo et al., {\sl Dual string with U(1) colorsymmetry},
   \NP{B111} (1976) 77.\\
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\nonfrenchspacing
\end{thebibliography}



%12345678901234567890123456789012345678901234567890123456789012345678901234567890
\chapter[Conformal Field Theory and OPEs]{Conformal Field Theory and
Operator Product Expansions\label{chp:CFT&OPEs}}
\mychapter
%\setcounter{page}{1}
This chapter gives an introduction to conformal field theory with
special attention to Operator Product Expansions (OPEs). It is of
course not complete as conformal field theory is a very wide subject, and
many excellent reviews exist, \eg \cite{Ginsparg,Itzykson,StAubin}.
Because it forms an introduction to the subject, some points are probably
trivial for someone who feels at home in conformal field theory. However,
some topics are presented from a new standpoint, a few new results (on
the associativity of OPEs) are given, and  notations are fixed
for the rest of the work.

We start by introducing the conformal transformations. Then, we study the
consequences of a symmetry of a conformal field theory on its correlation
functions. In the quantum case, this information is contained in the Ward
identities. These identities are then used in the third section to
develop the OPE formalism. We study the consistency conditions for OPEs in
detail and introduce the notion of an Operator Product Algebra (OPA). We then
draw attention to the close analogy between OPEs and Poisson brackets.
Section \ref{sct:modeAlgebra} defines the mode algebra of the symmetry
generators. In the next section, we define the generating functionals of the
theory and show how they are determined by the Ward identities.
We conclude with some important examples of conformal field
theories, free field theories and \WZW s.

\section{Conformal transformations \label{sct:confTrans}}
A conformal field theory is a field theory which is invariant under
general coordinate transformations $x^i \rightarrow f^i(x)$ and under
the additional symmetry of Weyl invariance. The latter transformations
correspond to local scale transformations of the metric
$g_{ij}(x)\rightarrow \Lambda(x) g_{ij}(x)$ and fields
$\Phi(x)\rightarrow \Lambda(x)^h \Phi(x)$, where $h$ is the scaling
dimension of the field $\Phi$. The combination of these symmetries can
be used to gauge away components of the metric. In two dimensions
one has exactly enough parameters to put the metric equal, at least
locally, to the Minkowski metric $\eta_{ij}$ (the conformal gauge).
Obviously, conformal transformations (coordinate transformations which
scale the metric) combined with the appropriate Weyl rescaling form a
residual symmetry group. Therefore, we will study this conformal group
first.


Conformal transformations are coordinate transformations which change the
metric with a local scale factor. In a space-time of signature $(p, q)$
they form a group isomorphic with $SO(p+1,q+1)$. However, in the complex
plane it is well-known that all (anti-)analytic transformations are
conformal. This extends to the Minkowski plane where in
light-cone coordinates, the conformal transformations are given by\col
\be
x^\pm \longrightarrow x'^\pm(x^\pm)\,.
\ee
In a space-time of signature $(-1,1)$, it is customary to perform a Wick
rotation. We will always assume this has been done, and treat only the
Euclidean case.

For a space of Euclidean signature, it is advantageous to use a complex
basis $(\tau+ i \sigma, \tau-i \sigma)$. In string theory, the space in
which these coordinates live is a cylinder, as $\sigma$ is used as a
periodic coordinate. This also applies to two-dimensional statistical
systems with periodic boundary conditions in one dimension. One then maps
this cylinder to the full complex plane with coordinates\be
(z, \zb) =
(\exp (\tau+ i \sigma), \exp (\tau- i \sigma)\,,\label{eq:cylplane}
\ee
where we will take a flat metric proportional to $\delta_{ij}$ in the real
coordinates, or in the complex coordinates\col
\be
ds^2 = 2\sqrt{g} dz d\zb\,.\label{eq:metric}
\ee
We will use the notation $x$ for a coordinate of a point in the complex
plane. Note that the points with fixed time $\tau$ lie on a circle
in the $(z, \zb)$ plane.

It is often convenient to consider $z$
and $\zb$ as independent coordinates (\ie not necessarily complex
conjugate). We can then restrict to the Euclidean plane by imposing
$\bar{z}=z^*$. The plane with a Minkowski metric corresponds to
$z,\bar{z}\in\Bbb{R}$.

As we are working in Euclidean space, the complex plane can be
compactified to the Riemann sphere. Conformal field theory can also be
defined on arbitrary Riemann surfaces, but we will restrict ourselves in
this work to the sphere.\puzzle{refs}


In the complex coordinates, a conformal transformation is given by\col
\bea
z \rightarrow z'=f(z), &&
\zb \rightarrow \zb'=\bar{f}(\zb) \, ,\label{eq:ConfTransf}
\eea
where $f$ is an analytic function, and $\bar{f}$ is antianalytic.
\begin{definition}\label{def:primary}
A primary field transforms under the conformal transformation
(\ref{eq:ConfTransf}) as\col
\bea
\Phi(z,\zb) &\rightarrow&  \Phi'(f(z),\bar{f}(\zb))  =
    (\dz f(z))^{-h} (\db \bar{f}(\zb))^{-\hb} \Phi(z,\zb)\, ,
\label{eq:PrimaryTransf}
\eea
where $\dz \equiv {d \over dz}$ and $\db \equiv {d \over d\zb}$. The numbers
$h$ and $\bar{h}$ are called the conformal dimensions of the field $\Phi$.
\end{definition}
For infinitesimal transformations of the coordinates $f(z) = z - \e(z)$, we
see that the primary fields transform as\col
\be
\delta_{\e} \Phi(z,\zb) =
   \e(z) \dz \Phi(z,\zb) + h \dz \e(z) \Phi(z,\zb)
\, . \label{eq:PrimaryTransfInf}
\ee
By choosing for $\e(z)$ any power of $z$ we see that the conformal
transformations form an infinite algebra generated by\col
\bea
l_m = -z^{m+1} \dz\,\, \mbox{ and }\,\,
\bar{l}_m = -\zb^{m+1} \db,&& m \in \Bbb{Z}\, ,
\eea
which consists of two commuting copies of the Virasoro algebra\puzzle{
\cite{VirasoroAlgebra}}, but without central extension (see further)\col
\be
 [l_m,l_n] = (m-n) l_{m+n} \,,\label{eq:ClassicalVir} \ee
and analogous commutators for the $\bar{l}_m$.

Clearly, $l_0$ corresponds to scaling transformations in $z$.
The combination $l_0+\bar{l}_0$ generates scaling transformations in the
complex plane $x\rightarrow \lambda x$, while $i(l_0-\bar{l}_0)$
generates rotations. This means that a field $\Phi(x)$ with conformal
dimensions $h$ and $\bar{h}$ has scaling dimension $h+\bar{h}$ and spin
$|h-\bar{h}|$.
$l_{-1}$ generates translations and $l_1$ ``special" conformal
transformations. The subalgebra formed by $\{l_{-1},l_0,l_1\}$  corresponds
to the globally defined and invertible conformal transformations\col
\be
z \rightarrow {a z+b \over c z +d} \, ,\label{eq:Mobius}
\ee
where $a, b, c, d \in \Bbb{C}$ and $ac-bd=1$.
These transformations form
a group isomorphic to $SL(2,\Bbb{C})\approx SO(3,1)$.\footnote{Although
there are two commuting copies of this algebra, we should take the
appropriate real section when we restrict to the real surface.}
\begin{definition}\label{def:quasiprimary}
A quasiprimary field transform as eqs.\
(\ref{eq:PrimaryTransf},\ref{eq:PrimaryTransfInf})
for the global transformations. A scaling field transforms this way
for translations and scaling transformations.
\end{definition}

\section{Correlation functions and symmetry}
The physics of a quantum field theory is contained in the correlation
functions. In a path integral formalism, a correlation function of
fields $\Phi_i$ corresponding to observables can be symbolically written
as\col
\be
< \car\{\Phi_1(x_1) \Phi_2(x_2)\cdots\} >\, =
{1\over\cn}\int [d\varphi_k]\,
      \exp (-S[\varphi_j])\, \, \Phi_1(x_1)\Phi_2(x_2) \cdots\,,
\label{eq:pathintcorr}
\ee
where $S$ is a suitable action, a functional of the fields $\varphi_j$ in the
theory, and $[d\varphi_k]$ denotes an appropriate measure. $\car$ denotes
time--order\-ing in a radial quantisation scheme \cite{FHJ:Rquant}, \ie
$|x_i|>|x_{i+1}|$. We will drop this symbol for ease of notation. In fact,
we will assume that for a correlation function $<\car\{\Phi_1(x_1)
\Phi_1(x_2)\ldots \}>$, the expression for $|x_1|<|x_2|$ can be obtained
by analytic continuation from the expression for $|x_2|<|x_1|$. This
property is commonly called ``crossing symmetry''. It can be checked for
free fields, and we restrict ourselves to theories where it is true.

The normalisation constant in \vgl{eq:pathintcorr} is given by\col
\be
\cn=\int [d\varphi_k]\, \exp (-S[\varphi_j])\,.\label{eq:normPathdef}
\ee
We suppose that $\cn$ is different from zero (vacuum normalised to one). In
general, the pathintegral \re{eq:pathintcorr} has to be computed
perturbatively.

Symmetries of the theory put certain restrictions on the form of the
correlation functions. We will investigate this in the next subsections.

\subsection{Global conformal invariance}
In this subsection, we consider the consequences of the invariance of the
correlation functions of quasiprimary fields under the global conformal
transformations
\re{eq:Mobius}. The results can be extended to conformal field theories in
an arbitrary number of dimensions, but we treat only the two-dimensional
case.

In conformal field theory one generally
requires translation, rotation and scaling invariance of the
correlation functions. This restricts all one-point functions of
fields with zero conformal dimensions ($h$ = $\bar{h} = 0$) to be
constant, and all others to be zero. Two-point functions have
the following form\col \be
<\Phi_1(z_1,\zb_1) \, \Phi_2(z_2,\zb_2)>\, = {{\rm cst} _{12} \over
(z_1-z_2)^{h_1+h_2}(\zb_1- \zb_2)^{\hb_1+\hb_2}} \, ,\label{eq:2point}
\ee
with ${\rm cst} _{12}$ a constant. If one requires global conformal
invariance (\ie invariance under the special transformations as well), only
fields with the same conformal dimensions can have non-zero two-point
functions, see also \vgl{eq:WardIdTij}.
Under this assumption, three-point functions are restricted to\col
\bea
\leqn{<\Phi_1(z_1,\zb_1) \, \Phi_2(z_2,\zb_2)\,\Phi_3(z_3,\zb_3)> =}
&&    {\rm cst}_{123} \prod_{\{ijk\}\atop{i<j}}{1 \over
        (z_i-z_j)^{h_{ijk}} (\zb_i-\zb_j)^{\hb_{ijk}} } \,  ,\label{eq:3point}
\eea
where $h_{ijk}=h_i+h_j-h_k$ and the product goes over all permutations of
$\{1,2,3\}$.\\
It is also possible to find the restrictions from global conformal
invariance on four (or more)-point functions. It turns out that the
correlation function is  a function of the harmonic ratios of the
coordinates involved, apart from $(z_i-z_j)$ factors like in
(\ref{eq:3point}) to have the correct scaling law. Crossing symmetry has
to be checked for a four-point function, while it is automatic for two-
and three-point functions.

Note that imposing invariance under the local conformal transformations would
make all correlation functions zero.

\subsection{Ward identities\label{ssct:WardId}}
In this subsection, we will discuss the consequences of a global symmetry of
the action using Ward identities.

We start by considering an infinitesimal transformation $\d_\e$ of the
fields, where $\e(x)$ is an infinitesimal parameter. Throughout this work,
we will only consider local transformations, \ie $\d_\e\Phi(x)$ depends on
a finite number of fields and their derivatives. If the transformation
leaves the measure of the pathintegral \re{eq:pathintcorr} invariant, we
find the following identity\col
\bea
\leqn{\int [d\varphi_k]\,\exp (-S[\varphi_j])\
   \d_\e\left( \Phi_1(x_1)\Phi_2(x_2) \cdots\right)\ =}
&&\int [d\varphi_k]\,\exp (-S[\varphi_j])\
   \left(\d_\e S[\varphi]\right) \Phi_1(x_1)\Phi_2(x_2) \cdots\,.
\label{eq:WIpre}
\eea
This follows by considering a change of variables in the pathintegral.
Using \vgl{eq:pathintcorr}, \vgl{eq:WIpre} implies an identity between
correlation functions. This Ward identity is useful in many cases, but at
this moment we are interested in transformations $\d_\e$ which leave the
action invariant if $\d_\e$ is constant, \ie global symmetries.

As an example, we will derive the Ward identity for the \emt\ $T^{ij}$.
Consider an infinitesimal transformation of the fields of the form\col
\be
\delta_{\e} \Phi(x) = \e^i(x) \partial_i \Phi(x) + \cdots
\, ,\label{eq:transPhi}
\ee
where the dots denote corrections according to the tensorial nature of the
field $\Phi$, see \re{eq:PrimaryTransfInf}. The action is supposed
to be invariant under these transformations if the $\e^i(x)$ are constants.
Using Noether's law, one has a classically conserved current associated to
this symmetry of the action $S$\col
\be
\d_{\e} S =  -{1\over \pi}\int d^2x\,\sqrt{g} T^i{}_j(x) \partial_i \e^j(x)\, ,
\label{eq:defTgenTrans}
\ee
where $g$ is the absolute value of the determinant of $g_{ij}$.\footnote{We
restrict ourselves to the case of a constant metric.}
This equation defines the \emt\ $T^{ij}$ up to functions
whose divergence vanishes. One can show that the alternative definition
\puzzle{kort bewijs}
\be
T_{ij} = {-2\pi\over \sqrt{g}} {\delta S \over \delta g^{ij}} \label{eq:defT}
\ee
satisfies (\ref{eq:defTgenTrans}). It is obvious that the tensor
\re{eq:defT} is symmetric. If the action is invariant under Weyl scaling, we
immediately see that it is traceless. Hence, in a conformal field theory in
two dimensions, the \emt\  has only two independent components.
In the complex basis, the components that remain are $T^{zz}$ and
$T^{\zb\zb}$. Using the metric \vgl{eq:metric}, we see from
\vgl{eq:defTgenTrans} that Weyl invariance implies that the action is
not only invariant under a global transformation, but also under a conformal
transformation, $\db\e^z=0, \e^\zb=0$. This transformation is sometimes
called semi-local.

Let us now consider the expectation value of some fields $\Phi_k$.
Combining eqs.\ \re{eq:WIpre} and \re{eq:defTgenTrans}, we find\col
\bea
\lefteqn{<\left(\delta_{\e}\Phi_1(x_1)\right) \Phi_2(x_2) \cdots>\,+\,
        <\Phi_1(x_1)\left(\delta_{\e} \Phi_2(x_2)\right) \cdots> +\cdots}
    \qquad\qquad \nonu
 &=& -{\sqrt{g}\over\pi} \int d^2x\,
         \left(\partial_i\e^j(x)\right) <T^i{}_j(x) \Phi_1(x_1)\Phi_2(x_2)
\cdots>\,. \label{eq:WardIdTij}
\eea
The equation \re{eq:WardIdTij} is the Ward identity for the \emt.
It shows that $T^{ij}$ is the generator of general coordinate
transformations. Note that $T^{ij}$ has to be symmetric and traceless for
the correlation functions to be rotation and scaling invariant.

We derived the Ward identity \re{eq:WardIdTij} in a formal way. In a given
theory, it has to be checked using a regularised calculation. We will not
do this in this work, and assume that \vgl{eq:WardIdTij} holds, possibly with
some quantum corrections. This enables
us to make general statements for every theory where \vgl{eq:WardIdTij} is
valid. Similar Ward identities can be derived for any global symmetry of
the action which leaves the measure invariant.

An important corrollary of the Ward identity for a symmetry generator, is
that it is conserved ``inside'' correlation functions. We will again treat
$T^{ij}$ as an example.  We take $\e^i(x)$ functions which go to zero when
$x$ goes to infinity, and which have no singularities. In this case,
we can use partial integration.
Now, the \lhs\ of (\ref{eq:WardIdTij}) depends only on the value of $\e$
(and its derivatives) in the $x_i$. This means that, after a partial
integration, the coefficient of $\e$ in the integrandum in the \rhs\
of (\ref{eq:WardIdTij}) has to be zero except at these points. We find\col
\be
{\partial\over\partial x^i} <T^{ij}(x) \Phi_1(x_1) \Phi_2(x_2)
\cdots>\, =\, 0\label{eq:eomTij}
\ee
where $x \neq x_k$.

From now on, we consider the case of a conformal field theory in two
dimensions. We already showed that $T^{ij}$ is symmetric and traceless,
in the sense that all correlation functions vanish if one of the fields
is the antisymmetric part of $T^{ij}$ or its trace. In the complex basis
the two remaining components are $T^{\zb \zb}$ and
$T^{zz}$. The conservation of $T^{ij}$ \vgl{eq:eomTij} gives\col
 \be
{d\over d\zb} <T^{\zb\zb}(z,\zb) \Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2) \cdots>
  = 0\,,\qquad z\neq z_i\, ,
\ee
and an analogous equation for $T^{zz}$. This shows that $<T^{\zb\zb}(z,\zb)
\Phi_1 \Phi_2\cdots>$ is a holomorphic function of $z$ with possible
singularities in $z_i$. Using the assumption that the
transformation of the fields $\d_\e\Phi_i(x_i)$ is local, we see that the
correlation function is a meromorphic function, with
possible poles in $z=z_i$. We define\col
\bea
T\equiv  { g} T^{\zb \zb}=T_{zz} &\mbox{and}&
\Tb\equiv { g}T^{zz}=T_{\zb\zb} \label{eq:TTbdef}
\eea
and we will write $T$ as a function of $z$ only.

For $\e^i(x)$ as specified above \vgl{eq:eomTij}, only the singularities
$z=z_i$ contribute to the Ward identity \re{eq:WardIdTij}. We now take
$\e^z=\e(z),\,\e^\zb=0$ in a
neighbourhood of these points. Using \vgl{eq:dbarToContour}, the Ward
identity \re{eq:WardIdTij} becomes for a conformal transformation\col
 \bea
\leqn{\sum_{j=1}^{N} <\Phi_1(z_1,\zb_1)\ \cdots\
      \left(\delta _{\e} \Phi_j(z_j,\zb_j)\right)\ \cdots > }
&=&  \oint_C {dz\over 2 \pi i}
            \e(z)< T(z) \Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >\, ,
 \label{eq:WardIdTcontour}
\eea
where the contour $C$ encircles all $z_j$ exactly once anticlockwise (and an
analogous formula for antianalytic variations).

If all $\Phi_j$ are primary fields, we can use (\ref{eq:PrimaryTransfInf}) to
compute the \lhs\ of the Ward identity (\ref{eq:WardIdTcontour})\col
\bea
\leqn{\oint_C {dz\over 2 \pi i}
        \e(z)< T(z) \Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots > }
&=&\sum_{j=1}^N \left(\e(z_j) \dz_j + h_j\, \dz_j \e(z_j)\right)
         <\Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >\,.
\label{eq:WardIdTPrimcontour}
\eea
This fixes the singular part in ($z = z_i$) of the meromorphic
$(N+1)$-point function $<\!T \Phi_1 \cdots\!>$.
We will assume that all
correlation functions have no singularity at infinity. Hence, the regular
terms in $(z-z_i)$ vanish, except for a possible constant.

To know the transformation law of $T(z)$ itself, we observe that since it is
classically a rank two conformal tensor, its dimensions are $h= 2, \,
\bar{h}=0$. However, due to quantum anomalies, a Schwinger term can arise in
its transformation law\col
\be
\delta_{\e} T(z) = \e(z) \dz T(z) + 2 \dz \e(z) T(z)+ {c\over12} \dz^3
\e(z)\,. \label{eq:TTransfInf}
\ee
In the path integral formalism, the Schwinger term is non-zero if the measure
is not invariant under the symmetry transformation generated by $T$, see
\vgl{eq:WardIdTij}. From dimensional arguments, it follows that $c$ should
have dimension zero. In fact, $c$ is in general a complex number.
When $c$ is different from zero, $T$ is not a primary field, but only
quasiprimary, see definition \ref{def:quasiprimary}.\\
The transformation law \re{eq:TTransfInf} involves only $T$ and no other
components of the \emt. In the classical case, this is a direct consequence
of the conformal symmetry of the theory. We will assume that the same
property holds in the quantum case.

Recall that \vgl{eq:WardIdTPrimcontour} determines the correlation
function $<\!T \Phi_1 \cdots\!>$ up to a constant. When we
require invariance of the correlation function under scaling
transformations ($\e(z)\sim z$), this constant has to be zero. Under
these assumptions, the Ward identity completely fixes all correlation
functions with one of the fields equal to $T$ and all other fields
being primary\col
\bea
\leqn{< T(z)
\Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >=} && \sum_{j=1}^N \left({ h_j\over
(z-z_j)^2} + {1\over (z-z_j)^{}} \dz_j \right)
         <\Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >\,.
\label{eq:WardIdTPhiPhiseries}
\eea

We can now look at correlation functions of $T$ only. Using
\vgl{eq:TTransfInf}, we find in a completely analogous way to
\vgl{eq:WardIdTPhiPhiseries}\col
\bea
\leqn{< T(z) T(z_1) T(z_2)\cdots >=}
\!\!\!\!\!\!&\! &\!\!\!\!\!\! \sum_{j=1}^N <\!T(z_1)\cdots\left(
        {c/2\over (z-z_j)^4} +{ 2T(z_j)\over (z-z_j)^2} +
            {\partial_j T(z_j)\over (z-z_j)^{}} \right)
          \cdots T(z_N)\!>\,.\label{eq:WardIdTTTseries}
\eea
Let us consider a few examples. Because of scaling invariance,
one-point functions of fields with non-zero dimension vanish. Together
with the fact that $<\unity>\,=\,1$, we see that \vgl{eq:TTransfInf}
implies\col \be
<\delta_\e T> = {c\over12} \dz^3 \e(z)\,.
\ee
\puzzle{anomaly of diffeomorphisms}
The Ward identity \re{eq:WardIdTcontour} fixes then the two-point function
to\col
\be
<T(z) T(w)>\ =\ {c/2 \over (z-w)^4}\,,
\ee
in accordance with \vgl{eq:2point}. Similarly, the constant in the
three-point function \re{eq:3point} of three $T$'s is fixed to $c$.

The analysis of the Ward identity for the \emt\ can be repeated for every
generator of a global symmetry of the theory. In two dimensions, all
tensors can be decomposed in symmetric tensors. If they are in addition
traceless, only two components remain $W^{zzz\cdots}$ and
$W^{\zb\zb\zb\cdots}$. We again find that $\db
W^{\zb\zb\zb\cdots}=0$, and the correlation functions of
$W^{\zb\zb\zb\cdots}$ are meromorphic functions. For generators
with a spinorial character, or generators with non-integer conformal
dimensions, we can only conclude that the correlation functions are
holomorphic, \ie fractional powers of $z-z_i$ could occur. We restrict
ourselves in this work to the meromorphic sector of the conformal field
theory. The symmetry algebra of the theory is a direct
product\footnote{This is only true when $z$ and $\zb$ are regarded is
independent coordinates.} $\ca\otimes\bar{\ca}$ where $\ca$ is the algebra
formed by the ``chiral'' generators (with holomorphic correlation
functions), and $\bar\ca$ is its antichiral counterpart. From now on, we
will concentrate on the {\sl chiral} generators. \puzzle{commutation of T Tb}

To conclude this section, we wish to stress that the Ward identities are
regularisation dependent. In this work, we will not address the question
of deriving the Ward identities for a given theory. However, once they
have been determined, the Ward identities enable us to compute the singular
part of any correlation function containing a symmetry generator in terms of
(differential polynomials of) correlation functions of the fields of the
theory. This determines those correlation functions up to a
constant, because we required that a correlation function has no
singularity at infinity. Furthermore, if the correlation functions are
invariant under scale transformations, this extra constant can only appear
when the sum of the conformal dimensions of all fields in the correlator is
zero.


\section{Operator Product Expansions\label{sct:OPEs}}
In this section, we discuss Operator Product Expansions (OPEs).
In a first step, we show how they encode the
information contained in the Ward identities. As such, OPEs provide an
algebraic way of computing correlation functions. In the next subsection,
we derive the consistency conditions on the OPEs. Subsection
\ref{ssct:OPAs} introduces the concept of an Operator Product Algebra
(OPA).

\subsection{OPEs and Ward identities\label{ssct:OPE&WI}}
To every field in a conformal field theory, we assign a unique element of a
vectorspace $\cv$ of ``operators''. For the moment, we leave the precise
correspondence open, but our goal is to define a bilinear operation in the
vectorspace which enables us to compute correlation functions.
We simply write the same symbol for the field and the corresponding
operator. Similarly, if a field has conformal dimension $h$, we say that
the corresponding operator has conformal dimension $h$.

Let us look at an example to clarify what we have in mind.
Consider the Ward identities of the chiral component of the \emt\ $T(z)$.
Eq.\ \re{eq:WardIdTPhiPhiseries} suggests that we assign the OPE\col
\be
T(z) \Phi(z_0,\zb_0) \,=\, {h \Phi(z_0,\zb_0)\over(z-z_0)^2} +
      {\dz\Phi(z_0,\zb_0)\over z-z_0^{}} + O(z-z_0)^0
\label{eq:PrimaryFieldOPE}
\ee
for a primary field $\Phi$ with conformal dimension $h$. We do not
specify the regular part yet. Similarly,
\vgl{eq:WardIdTTTseries} imposes the following OPE for $T$ with itself\col
 \be
T(z) T(z_0) \,=\,
{c/2\over(z-z_0)^4}+{2 T(z_0)\over(z-z_0)^2} + {\dz T(z_0)\over z-z_0^{}} +
O(z-z_0)^0\,. \label{eq:VirasoroOPE}
\ee

We see that an OPE is a bilinear map from $\cv \otimes \cv$ to the
space of formal Laurent expansions in $\cv$. We will use the following
notation for OPEs\col
\be T_1(z) T_2(z_0) =
     \sum_{n <= h(T_1,T_2)}{\frac{{[}T_1T_2{]}_n(z_0)}{(z-z_0)^n}} \, ,
\label{eq:OPEdef}
\ee
where $h(T_1,T_2)$ is some finite number. Because of the correspondence
between OPEs and Ward identities, we see that  all terms in the sum
\re{eq:OPEdef} have the same conformal dimension\col\ ${\rm dim}([T_1T_2]_n)
= {\rm dim}(T_1) + {\rm dim}(T_2) - n$. If no negative dimension fields are
present in the field theory, we immediately infer that $h(T_1,T_2)$ is less
than or equal to the sum of the conformal dimensions of $T_1$ and $T_2$.
As noted before, we restrict ourselves in this work to the case
where the correlation functions of the chiral symmetry
generators $T_i(z_i)$ are meromorphic functions in $z_i$. In other words,
the sum in \vgl{eq:OPEdef} runs over the integer numbers. We denote the
singular part of an OPE by $ T_1\SING{(z)T_2}(z_0)$.
It is determined via\col
\be
\oint_{C_{z_0}} {dz\over 2\pi i}\e(z) T_1\SING{(z)T_2}(z_0) =
\d^{T_1}_\e T_2(z_0)\,,\label{eq:OPEsingdef}
\ee
where we denoted the transformation generated by $T_1$ with parameter
$\e$ as $\d^{T_1}_\e$. Note that we can assign an OPE only when $T_1$ is a
symmetry generator.

The prescription we use for the moment to calculate correlation
functions with OPEs, is simply an application of conformal Ward
identities like \vgl{eq:WardIdTcontour}. As an example, to compute a
correlation function with a symmetry generator $T_1$, we substitute
the contractions of $T_1$ with the other fields in the
correlator\footnote{In fact, this is only valid when the correlation
function is zero at infinity, \ie when no extra constant appears, see
the discussion at the end of the previous section.}\col
\bea
\lefteqn{<T_1(z)  \Phi_1(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >\ =}
\qquad\nonu &&<T_1\SING{(z) \Phi_1}(z_1,\zb_1) \Phi_2(z_2,\zb_2)\cdots >+\nonu
&&<T_1\SING{(z) \Phi_1(z_1,\zb_1) \Phi_2}(z_2,\zb_2)\cdots >+\cdots
\label{eq:corOPEcontraction} \eea
In this way, we can always reduce any correlation function, where
symmetry generators are present to a differential polynomial of correlation
functions of fields only. Of course, if only generators were present from
the start, we will end up with a function of all arguments $z,w,\ldots$
containing one-point functions and $<\unity >=1$.

We now consider the map from the space of fields to the vectorspace
$\cv$ in more detail. The normal ordered product of two fields $T_1$ and
$T_2$ is defined by considering a correlation function $<T_1(z) T_2(w)
\,X>$, where $X$ denotes an arbitrary sequence of operators, and taking the
limit of $z$ going to $w$ after substracting all singular terms\col
 \bea \leqn{< :T_1 T_2:(w) \,X > = }
&&\lim_{z\rightarrow w} <\left(T_1(z) T_2(w)-
          \sum_{n>0} {[T_1T_2]_n(w)\over(z-w)^n}\right) \,X >\,.
\label{eq:pointsplreg}
\eea
This definition corresponds to a point-splitting regularisation
prescription. In the case of non-meromorphic correlation functions, extra
fractional powers of $(z-w)$ are included in this definition. We will not
treat this case here. In the notation of \vgl{eq:OPEdef}, we see that
the map from the space of the fields to $\cv$ is in this case
given by\col  \be
:T_1T_2:\rightarrow[T_1 T_2]_0 \,.\label{eq:NOdef}
\ee
In fact, we can use the same procedure to define all operators in the regular
part of the OPE\footnote{It is sufficient to define an operator $A$ by
specifying all correlators $<A(z)\,X>$. Crossing symmetry of the
correlators gives then $<X\,A(z)\,Y>$, where $X$ and $Y$ stand for arbitrary
sequences of operators.}, \ie
\bea
\leqn{< \sum_{n\leq 0} {[T_1T_2]_n(w)\over(z-w)^n} \,X >\ =\hspace*{3em} }
&& <\left(T_1(z) T_2(w)- \sum_{n>0} {[T_1T_2]_n(w)\over(z-w)^n}\right)
\,X >\,.\label{eq:OPEregdef}
\eea
The definition \re{eq:OPEregdef} implies that correlation
functions can be computed by substituting for two operators their {\sl
complete} OPE. We write\col \be
<T_1(z_1)T_2(z_2)\ X> =\ <\FULL{T_1(z_1)T_2(z_2)}\ X>
\label{eq:corOPEreg}\,.
\ee
In this way, an OPE is now an identity between two bilocal operators.

Because one-point functions are constants and we assumed that correlators
have no singularity at infinity, the definition \re{eq:OPEregdef}
implies\col
\bea
 <[T_1(z) T_2(w)]_n >\ =\ 0\,,&&\ \forall n<0\,. \label{eq:corregzero}
\eea

If all fields in the correlation function \vgl{eq:corOPEreg} are symmetry
generators, we end up with one-point functions $<T_i(z)>$. These can of
course not be computed in the OPE formalism. As infinite sums are involved,
one should pay attention to convergence, which will in general only be
assured when certain inequalities are obeyed involving the $z_i$. However,
the result should be a meromorphic function with poles in $(z_i-z_j)$. So
analytic continuation can be used.

The definition \re{eq:OPEregdef} gives the same
results as the straightforward application of the Ward identities
\vgl{eq:corOPEcontraction}. This will be proved in the next subsection
after determining the consistency requirements on the OPEs.

\subsection{Consistency conditions for OPEs \label{ssct:OPErules}}
In this subsection, we determine the consistency requirements on the OPEs
by considering (contour integrals of) correlators. We first list
the properties of correlation functions used in subsection
\ref{ssct:WardId}\col
\begin{assumption}\label{ass:corrfunctions}
The correlators have the following properties\col
\begin{itemize}
\item translation and scaling invariance
\item no singularity at infinity
\item the correlation functions involving a chiral symmetry generator
$W(z)$ are meromorphic functions in $z$
\item crossing symmetry
\end{itemize}
\end{assumption}

We now suppose that there is a map from the space of fields of the
conformal field theory to a vectorspace $\cv$. A $\Bbb{Z}_2$ grading in
$\cv$ should exist, corresponding to bosonic and fermionic operators. We
denote it with $|A|$.
There is an even linear map $\del$ from $\cv$ to $\cv$, and a sequence of
bilinear operations for every $n\in\Bbb{Z}$ which we denote by $[AB]_n$,
see \vgl{eq:OPEdef}. For $n>0$ the bilinear operations are defined in
\vgl{eq:OPEsingdef}. We require that correlation functions can be
computed by substituting the complete OPE as done for \vgl{eq:corOPEreg}.
From this definition, we determine the properties of these maps. In the
remainder of this subsection $X$ and $Y$ denote arbitrary sequences of
operators.

We first determine $[\del A\ B]_n$ by requiring that $\del$ corresponds to
the derivative on the fields. Because we have\col
\be
<X\ \del A(z)\ Y> = {d\over dz} <X\ A(z) Y>\,,
\ee
we find that the OPE $\del A(z)B(w)$ is given by taking the derivative of
the OPE $A(z)B(w)$ with respect to $z$\col \be
\dz A(z)B(w) =
       \sum_{n <= h(A,B)}{-n\frac{[AB]_n(w)}{(z-w)^{n+1}}} \,
,\label{eq:OPEderL} \ee
and hence\col
\be [\dz A\,B]_{n+1} = -n [AB]_n \label{eq:PolederL} \,.
\ee
This equation enables us to write all the terms in the regular part
in terms of normal ordered operators\col
\be
[AB]_{-n} = {1\over n!} :(\dz^n \!\! A)\ B:\,,\,\,\,\,\,n\in\Bbb{N}\,.
\label{eq:Poleregdef}
\ee
By applying derivatives with respect to $w$ we find\col
\be
[A\,\dz B]_{n+1} = n [AB]_n + \dz [AB]_{n+1}\label{eq:PolederR} \,.
\ee

We now investigate the consequences of the crossing symmetry of the
correlators.

First, consider a correlator $<A(z)B(w)\,X>$. As $X$ is completely
arbitrary, we see that the OPE $A(z) B(w)$ must be equal to the OPE
$(-1)^{|A|\,|B|} B(w) A(z)$. Looking at \vgl{eq:OPEdef} for both OPEs, one sees
that a Taylor expansion of $[AB]_n(z)$ around $w$ has to be performed to
compare the OPEs. The result is\col
\be
 {[}BA{]}_q = (-1)^{|A|\,|B|}
\sum_{l\geq q}{\frac{(-1)^l}{(l-q)! } \partial^{(l-q)}{[}AB{]}_l}
\label{eq:OPEcomm}
\ee
for arbitrary $q$.
\begin{intermezzo}\label{int:OPEAA}%
Eq.\ \re{eq:OPEcomm} leads to a consistency equation when $A=B$\col
\be
 [AA]_q = -\sum_{l>0}{(-1)^{l}\over 2l!} \partial^l [AA]_{q+l}
\,\,\,\,
{\rm if }|A|+ q\,\, {\rm odd}\,, \label{eq:OPEcommAA}
\ee
where we shifted the summation index $l$ over $q$ with respect to
\vgl{eq:OPEcomm}. If $A$ is bosonic (fermionic), this relation determines an
odd (even) pole of the OPE $A(z)A(w)$ in terms of derivatives of the higher
poles, and thus in terms of the higher even (odd) poles. We can write\col
\[
 [AA]_q =
\sum_{k\geq 0}a_k \partial^{2k+1}{[}AA{]}_{q+2k+1}\,,
\]
where the constants $a_k$ satisfy the following recursion relation\col
\[
a_k = {1\over 2}\left({1\over (2k+1)!}-
         \sum_{0\leq l<k} {a_l\over (2k-2l)!}\right)\,,
\]
which can be solved by using the generating function\col
\[
f(x)=\sum_{k\geq 0} a_k \, x^{2k} = {\tanh(x/2)\over x}\,,
\]
for which the Taylor expansion is well known. Hence\col
\[
a_k = 2 B_{2k+2} {2^{2k+2}-1 \over (2k+2)!}\,,
\]
where $B_n$ are the Bernoulli numbers. The first values in this series
are\col
\[
a_0={1\over 2}\,,\,\,a_1=-{1\over 24}\,,\,\,a_2={1\over 240}\,,\,\,
a_3=-{17\over 40320}\,.
\]
Applying the rule \re{eq:OPEcommAA} for $q=0$ and $A$
fermionic, expresses the normal ordered product of a fermionic field with
itself in terms of the operators occuring in the singular part of its OPE
with itself. An example is found in the $N=1$ superconformal algebra. This
algebra contains a supersymmetry generator $G$ which has the following
OPE\col
\[
G(z)G(w)={2c/3\over(z-w)^3}+{2T(w)\over(z-w)} + O(z-w)^0\,.
\]
Applying the above formulas, we find\col
\[
{[}GG{]}_0 =\partial T\,.
\]
\end{intermezzo}

It is clear that \vgl{eq:OPEcomm} shows that (dropping sign
factors)\col
\bea
\leqn{<\FULL{\FULL{A(z)B(w)}C(u)}\,X>_{\rm OPE}}
&=&<\FULL{\FULL{B(w)A(z)}C(u)}\,X>_{\rm OPE}\nonu
&=&<\FULL{C(u)\FULL{A(z)B(w)}}\,X>_{\rm OPE}\,.
\eea
This does not yet prove that the correlator is crossing symmetric. Indeed,
for this we also need (dropping arguments as well)\col
\be
<[[AB]C]\ X>_{\rm OPE} \ = \ <[[CA]B]\ X>_{\rm OPE}\,.
\ee
We see that crossing ssymmetry implies ``associativity''  of the OPEs\col\
the order in which the OPEs are substituted should be irrelevant. This puts
very stringent conditions on the OPEs. These conditions are most easily
derived by using contour integrals of correlators. Indeed, we can isolate
the contribution of a certain part of the OPE by taking appropriate
contour integrals\col
\bea \leqn{<[A[BC]_p]_q (u) \,X>\, =
 \oint_{C_u} {dz\over 2\pi i}\, (z-u)^{q-1}\hspace*{3em}}
&&        \oint_{C_u} {dw\over 2\pi i}\, (w-u)^{p-1}\,<A(z)B(w)C(u) \,X>\,,
\label{eq:OPEcontoursR}
\eea
where $C_u$ denotes a contour which encircles $u$ once anti-clockwise, not
including any other points involved in the correlator. We can now use
a contour deformation argument relating the contour integral
in \vgl{eq:OPEcontoursR} to a contour integral where the integration over
$w$ is performed last, see fig.\ \ref{fig:contours}. This integral has two
terms\col\ one where the $z$ contour is around $u$ (corresponding to the
correlator $(-1)^{|A|\,|B|}<\FULL{B\FULL{AC}}\, X>$, and one where it is
around $w$ ($<\FULL{\FULL{AB}C}\,X>$).
\begin{figure}[thb]
\begin{center}
\portpict{cont}(9,3.2)
\end{center}
\caption{Contour deformations \label{fig:contours}}
\end{figure}
Using the definition \re{eq:OPEdef} for the OPEs, and Cauchy's residue
formula for contour integrals, we arrive at\col
\bea
[A[BC]_p]_q
=(-1)^{|A|\,|B|} [B[AC]_q]_p + \sum_{l>0}\bin{q-1}{l-1}[[AB]_{l}C]_{p+q-l}
    \label{eq:OPEJacRAB}\,.
\eea
A second equation follows in the same manner by interchanging the role of
$z$ and $w$ in fig.\ \ref{fig:contours} and bringing the second contour
integral of the \rhs\ of eq.\ \re{eq:OPEcontoursR} to the left\col
\bea
\leqn{[A[BC]_p]_q }
&=&(-1)^{|A|\,|B|}\left( [B[AC]_q]_p -
       \sum_{l>0}\bin{p-1}{l-1} [[BA]_{l}C]_{p+q-l}\right)\,.
    \label{eq:OPEJacRBA}
\eea
Both eqs.\ \re{eq:OPEJacRAB},\re{eq:OPEJacRBA} have to be satisfied for
any $p,q\in\Bbb{Z}$.\footnote{See appendix \ref{app:Combinatorics} for
the extended definition of the binomial coefficients.}

Similarly, by starting with\col
\bea
\leqn{<[[AB]_p\ C]_q (u) \,X>\, =
\oint_{C_u} {dw\over 2\pi i}\, (w-u)^{q-1}}
&&        \oint_{C_w} {dz\over 2\pi i}\, (z-w)^{p-1}\,<A(z)B(w)C(u) \,X>\,,
\eea
and using the same contour deformation of fig.\ \ref{fig:contours}, but now
with the first term of the \rhs\ brought to the left, we get\col
\bea [[AB]_p\ C]_q &=&
    \sum_{l\geq q} (-1)^{q-l} \bin{p-1}{l-q}[A[BC]_l]_{p+q-l} -
\label{eq:OPEJacL}\\
  && (-1)^{|A|\,|B|} \sum_{l>0} (-1)^{p-l} \bin{p-1}{l-1}[B[AC]_{l}]_{p+q-l}\,.
\nonumber
\eea
which has again to be satisfied for all $p,q\in \Bbb{Z}$.

Eq.\ \re{eq:OPEcomm} and \vgl{eq:OPEJacRAB} for
$q=0$ first appeared in \cite{bbss} where they were derived using the mode
algebra associated to OPEs (see section \ref{sct:modeAlgebra}). In
\cite{stps}, contour deformation arguments were used to find
\vgl{eq:OPEJacRAB} and an equation related to \vgl{eq:OPEJacRBA} for
restricted ranges of $p,q$. Reference \cite{stps} also contains
\vgl{eq:OPEJacL} for $q>1$.

At this point, we found the conditions on the OPEs of the operators in $\cv$
such that we find ``correlation functions'' which satisfy the assumptions
\ref{ass:corrfunctions}. We still need to show that this procedure indeed
gives the correlation functions of the conformal field theory we started
with, \ie that we find correlation functions which satisfy the Ward
identities of the theory. Before proceeding, we prove the
following lemma\col
\begin{lemma}\label{lemma:OPEJac}
The associativity condition \vgl{eq:OPEJacRAB} for strictly positive $q$ can
be rewritten as\col
%FULL??
\[
A\SING{(z) [B(w)}C(u)] = (-1)^{|A|\,|B|}\FULL{B(w)A\SING{(z)C}(u)} +
\FULL{A\SING{(z) B}(w)C(u)}
\]
for $|w-u|<|z-u|$.
\end{lemma}
\begin{proof}
The proof of this lemma is in some sense the reverse of the derivation of
\vgl{eq:OPEJacRAB}. We multiply \vgl{eq:OPEJacRAB} with $(z-u)^q(w-u)^p$
and sum for $p$ and $q$ over the appropriate ranges. After substracting
the result from the proposition of this lemma, we need to prove\col
\beastar
\leqn{ \FULL{A\SING{(z) B}(w)C(u)}\ =}
&&\sum_{q>0}\sum_p
(z-u)^{-q} (w-u)^{-p}\sum_{l>0}\bin{q-1}{l-1}[[AB]_lC]_{p+q-l}(u)\,.
\eeastar
Due to the binomial coefficient, and because $q$ is strictly positive, the sum
over $l$ is from one to $q$. Call $r=p+q-l$ and $s=q-l$. We find\col
\beastar
\rhs
&=&
   \sum_{r}\sum_{l>0}[[AB]_lC]_r(u)\\
&&\hspace*{2.5em}\left(
      \sum_{s\geq 0}\bin{l+s-1}{l-1}(z-u)^{-l-s}(w-u)^{-r+s}
   \right)\nonu
&=&\sum_{r}\sum_{l>0}
      (z-u)^{-l}(w-u)^{-r} (1-{w-u\over z-u})^{-l}
   [[AB]_lC]_r(u)\,,
\eeastar
which is exactly the \lhs. In the last step we used the Taylor expansion of
$(1-x)^{-l}$ which only converges for $|x|<1$.
\end{proof}
The Ward identities have the form \re{eq:WardIdTcontour}\col
\bea
\leqn{ \oint_C {dz\over 2 \pi i}
            \e(z)<\FULL{T_0(z) T_1(z_1)} \cdots T_n(z_n)> =}
&&\oint_{C_1} {dz\over 2 \pi i}\e(z)<T_0\SING{(z) T_1}(z_1) \cdots
T_n(z_n)> + \cdots\nonumber\\
&& +
 \oint_{C_n}{dz\over 2 \pi i}\e(z) <T_0\SING{(z) T_1(z_1) \cdots
      T_n}(z_n)>\,, \label{eq:WardIdgencontour}
\eea
where the contractions correspond by definition to the transformation
generated by $T_0$ of the field $T_i$, \vgl{eq:OPEsingdef}. The contour $C$
surrounds $z_1\cdots z_n$ once in the anti-clockwise direction, while the
contours $C_i$ encircle only $z_i$. $\e(z)$ is analytic in a region
containing $C$.
\begin{theorem}\label{trm:fullsing}
The Ward identities \vgl{eq:WardIdgencontour} are satisfied for correlation
functions which we compute by substituting two operators with their
complete OPE.
\end{theorem}
\begin{proof}
The $(n+1)$-point function $<T_0(z) T_1(z_1) \cdots T_n(z_n)>$ has only
poles in $z$ when $z=z_i$. We will use this to deform the contour $C$.\\
Suppose that the theorem holds for $n$-point functions. We rearrange the
operators $T_i$ such that $|z_1-z_2|$ is smaller than all other
$|z_i-z_j|$. We substitute $T_1(z_1)T_2(z_2)$ by their OPE. Then we have to
compute a $n$-point function.

We now split the contour $C$ in a contour
$C_2'$ and the contours $C_i$, $i>2$, used in \vgl{eq:WardIdgencontour}.
$C_2'$ encircles $z_1$ and $z_2$, but
no other $z_i$. Because of the rearrangement, we can take $C_2'$ such
that for any point $z$ on $C_2'$, $|z-z_2|>|z_1-z_2|$. By applying the
recursion assumption we find\col
\beastar
\leqn{ \oint_C {dz\over 2 \pi i}
            \e(z)<T_0(z) T_1(z_1) \cdots T_n(z_n)> =}
&&  \oint_{C_2'} {dz\over 2 \pi i}
            \e(z)<T_0\SING{(z) [T_1(z_1)}T_2(z_2)] \cdots T_n(z_n)>\\
&&+  \oint_{C_3} {dz\over 2 \pi i}
            \e(z)<T_0\SING{(z) [T_1(z_1)T_2(z_2)]T_3}(z_3) \cdots T_n(z_n)>
+ \cdots
\eeastar
We can now use lemma \ref{lemma:OPEJac} for the first term of the \rhs. We
indeed find the \rhs\ of \vgl{eq:WardIdgencontour}.
\end{proof}

We comment on when a correlation
function can be computed by taking all contractions, which was the
prescription we used to define OPEs in the previous subsection, see
\vgl{eq:corOPEcontraction}. Computing a correlation function in this way,
means that we drop the integrals in the Ward identity
\vgl{eq:WardIdgencontour}. This can only be done if all one-point functions
$<[T_1T_2]_0>$ vanish (except if $T_i=\unity$). Because of
scaling invariance, this is true when all
operators (except $\unity$) have strictly positive dimension.\\
Bowcock argued in \cite{Bowcockcontraction} that correlation functions
can be computed by substituting the complete OPE, or by using
contractions. His argument is based on the claim that
\vgl{eq:WardIdgencontour} is true, but no proof is given.

To conclude this subsection, we wish to mention that Wilson Operator Product
Expansions were already used outside the scope of two-dimensional conformal
field theory \cite{Wilson,Symanzik}. However, because the consistency
requirements on the OPEs are especially strong in two-dimensional conformal
field theory (due to the fact that the conformal algebra has infinite
dimension), it is there where the full power of the formalism comes to
fruition.

\subsection{Operator Product Algebras\label{ssct:OPAs}}
We can assemble the consistency conditions on OPEs in a definition,
see also \cite{Borcherds,Getzler:N=2}.
\begin{definition}\label{def:OPAnonull}
An Operator Product Algebra (OPA) is a $\Bbb{Z}_2$ graded vectorspace $\cv$
with elements $\unity,A,B,C\cdots$, an even linear map $\partial$, and a
bilinear binary operation\col
\beastar
[\,.,.\,]_l:\cv\otimes\cv \rightarrow \cv,&& l\in\Bbb{Z}\,,
\eeastar
which is zero for $l$ sufficiently large.
The following properties hold\col
\begin{itemize}
\item{\bf unity}: \[ [\unity A]_l = \delta_l\ A\]
\item{\bf commutation} :  (\vgl{eq:OPEcomm})\[
 {[}BA{]}_n = (-1)^{|A|\,|B|}
\sum_{l\geq n}{\frac{(-1)^l}{(l-n)! } \partial^{(l-n)}{[}AB{]}_l}
\hspace*{2.5em} \forall n\in \Bbb{Z}
\]
\item {\bf associativity} : (\vgl{eq:OPEJacRAB})
\beastar
[A[BC]_p]_q
&=&(-1)^{|A|\,|B|} [B[AC]_q]_p \\
&&+ \sum_{l>0}\bin{q-1}{l-1}[[AB]_{l}C]_{p+q-l}
\hspace*{2.5em} \forall p,q\in \Bbb{Z}
\eeastar
\end{itemize}
\end{definition}
The properties in this definition are sufficient to recover all consistency
conditions of the previous subsection. Using $\del A=[A\unity]_1$ and the
associativity condition, the equations for OPEs of derivatives
(\ref{eq:PolederL},\ref{eq:PolederR}) follow. Similarly, the other
associativity conditions \re{eq:OPEJacRBA} and \re{eq:OPEJacL} can be
derived from the properties of an OPA.

An alternative definition would be to impose eqs.\ (\ref{eq:PolederL},
\ref{eq:PolederR}), requiring the associativity condition
\vgl{eq:OPEJacRAB} only for $p, q\geq 0$. One then usually considers
\vgl{eq:OPEJacRAB} for $p$ or $q$ equal to zero as being the definition
of how we can calculate OPEs of normal ordered products. The set of
consistency conditions \vgl{eq:OPEJacRAB} for $p, q>0$ has then to be
checked for all generators of the OPA.

In some cases, the definition \ref{def:OPAnonull} is too strong.
OPEs are intended to compute correlation functions. It is possible that
there are some fields in the theory which have vanishing correlators with
all other fields. These null fields should be taken into account in the
definition of an OPA. Indeed, they could occur in every consistency equation
for OPEs without affecting the results for the correlation functions.
We need an algebraic definition of a null field. From the way we compute
correlators using OPEs, we see that if $N$ is a null field, $[NA]_n$ should
be again a null field, for any $A\in\cv$ and $n\in \Bbb{Z}$. If this would
not be true, we could write down a nonvanishing correlator with $N$. We see
that the null fields form an ideal in the OPA.
\begin{definition}\label{def:OPA}
Consider an OPA as defined in def. \ref{def:OPAnonull}. Suppose there is an
ideal $\cn$ in the OPA, whose elements we call null operators (or
null fields). We extend the definition of an OPA to algebras where the
defining properties are only satisfied up to elements of $\cn$.
\end{definition}
This extension of the definition of an OPA was
not considered in \cite{Borcherds,Getzler:N=2}.

It is in general difficult to check if we can consider an operator $N$ to
be null. We do not want that $\unity\in\cn$, because then all operators are null.
Hence, a necessary condition for $N$ to be null is that we can find no
operator $A$ in $\cv$ such that $[NA]_n \sim \unity$ (for some $n$).
Usually, this is regarded as a sufficient condition, because
one generally works with fields of strictly positive conformal dimension.
Due to the scaling invariance of the correlation functions, all one-point
functions are then zero, except $<\unity>$. So, in this case, any field which
does not produce the identity operator in some OPE has zero correlation
functions.

\subsection{OPA--terminology\label{ssct:OPAterms}}
In this subsection, we introduce some terminology which is continually
used in the rest of this work.

\begin{definition}\label{def:composite}
We call $A$ a {\sl composite operator} if it is equal to $[BC]_0$,
for some $B,C\neq\unity$, except when $B$ is equal to $C$ and fermionic.
\end{definition}
The condition when $B= C$ in this definition comes from the
considerations in intermezzo \ref{int:OPEAA}.

\begin{definition}
A set of operators is said to {\sl generate} the OPA if all elements of
$\cv$ can be constructed by using addition, scalar multiplication,
derivation and taking composites.
\end{definition}

\begin{definition}\label{def:Virasoro}
A {\sl Virasoro} operator $T$ has the following non-zero poles in its OPE
\re{eq:VirasoroOPE}\col
\[ [TT]_4={c\over2} \unity,\qquad [TT]_2= 2T,
\qquad [TT]_1=\del T\,. \]
\end{definition}

\begin{definition}\label{def:COPA}
An operator $A$ is a {\sl scaling} operator with
respect to $T$,  with (conformal) dimension $h$ if\col
\[ [TA]_2 = h A,\qquad [TA]_1=\del A\,.\]
A {\sl quasiprimary} operator is a scaling operator with in addition
$[TA]_3=0$.\\
A {\sl primary} operator is a scaling operator with in addition
$[TA]_n=0$ for $n\geq 3$.\\
A {\sl conformal OPA} is an OPA with a Virasoro $T$. All
other operators of the OPA (except $\unity$) are required to be scaling
operators with respect to $T$.
\end{definition}

\begin{intermezzo}\label{int:dimension}%
As an application of the above definitions, we wish to show that when $A$ and
$B$, with dimension $a$ and $b$, are scaling operators with respect to $T$
then $[AB]_n$ is a scaling operator with dimension  $a+b-n$.
Let us compute the first order pole of this operator with $T$ using
\vgl{eq:OPEJacRAB}\col
\begin{eqnarray*}
[T[AB]_n]_1 &=& [A[TB]_1]_n + [[TA]_1B]_n \\
&=&[A\ \partial B]_n + [\partial A\ B]_n\\
&=&\partial [AB]_n\,,
\end{eqnarray*}
where we used the sum of \vgl{eq:PolederL} and \vgl{eq:PolederR} in the last
step. For the second order pole we have\col
\begin{eqnarray*}
[T[AB]_n]_2 &=& [A[TB]_2]_n + [[TA]_2B]_n +[[TA]_1B]_{n+1}\\
&=&(a+b)[AB]_n + [\partial A\ B]_{n+1}\\
&=&(a+b-n)[AB]_n.
\end{eqnarray*}
\end{intermezzo}

\begin{definition}\label{def:dimension}
A map from the OPA to the half-integer numbers is
called a {\sl dimension} if it has the properties\col
\beastar
{\rm dim}(\unity)&=&0\\
{\rm dim}(\dz A)&=& {\rm dim}(A)+1\\
{\rm dim}([AB]_l)& =&{\rm dim}(A)+{\rm dim}(B)-l\,.
\eeastar
If such a map exists, we call the OPA {\sl graded}.
\end{definition}

\begin{definition}\label{def:WA}
A \WA\ is a conformal OPA where one can find a set of
generators which are quasiprimary.
\end{definition}
Different definitions of a \WA\ exist in the literature. Sometimes one
requires that the generators are primary (except $T$ itself). In this
work, we will mainly consider \WA s of this subclass, and for which the
number of generators is finite. The importance of \WA s lies in the fact
that the chiral symmetry generators of a conformal field theory form a
\WA\footnote{In fact, the chiral symmetry generators form only a conformal
OPA, but we know of no example in the literature where no quasiprimary
generators can be found.}. We will treat \WA s in more detail in chapter
\ref{chp:conformalInvariance}.

Finally, we introduce a notation for OPEs which lists only the operators in
the singular terms, starting with the highest order pole. As an example, we will
write a Virasoro OPE \re{eq:VirasoroOPE} as\col
\be
T\times T\ =\ \opefour{{c\over 2}}{0}{2T}{\partial T}\,.\label{eq:OPEnotation}
\ee

\subsection{Poisson brackets\label{ssct:Poisson}}
To conclude this section, we want to show the similarity between
Poisson bracket calculations and OPEs.

In a light-cone quantisation scheme (choosing $\zb$ as ``time'')
\cite{WZW2}, the symmetry generators in
classical conformal field theory obey Poisson brackets of the form\col
\be
\{A(z),B(z_0)\}_{\rm PB} = \sum_{n>0}{(-1)^{n-1} \over (n-1)!}\{AB\}_n(z_0)
    \dz^{n-1}\delta(z-z_0)\,,
\label{eq:PBdef}
\ee
where $\{AB\}_n$ are also symmetry generators of the theory.
The derivative is with respect to the $z$--coor\-di\-nate.
We choose the normalisation factors such that\col
\be
\{AB\}_n(z) =\int dz\ (z-w)^{n-1} \{A(z),B(w)\}_{\rm PB}\,.
\ee
For convenience, we drop the subscript {\rm PB} in the rest of this
subsection. The Poisson brackets satisfy\col
\bea
\{A(z),B(z_0)\}\!\!&=&\!\!(-1)^{|A|\,|B|} \{B(z_0),A(z)\}\nonu
\{\partial A(z),B(z_0)\}\!\!&=&\!\!{d\over dz}\{A(z),B(z_0)\}\nonu
\{\{A(z_1),B(z_2)\},C(z_3)\}\!\!&=&\!\!\{A(z_1),\{B(z_2),C(z_3)\}\}-\nonu
\!\!&&\!\! (-1)^{|A|\,|B|} \{B(z_2),\{A(z_1),C(z_3)\}\}
    \label{eq:PBproperties}
\eea
These relations imply identities for the $\{AB\}_n$. We do not
list the consequences of the first two, as they are exactly the same as
eqs.\ \re{eq:OPEcomm} and \re{eq:RulePolederL}. The Jacobi
identities give\col
\bea
\leqn{\{A\ \{BC\}_p\}_q(z_3)\ -\ (-1)^{|A|\,|B|}\{B\ \{AC\}_q\}_p(z_3)}
&=&\int  dz_2\ (z_2-z_3)^{p-1} \int dz_1\
   \sum_{l=1}^q \bin{q-1}{l-1}\nonu
&&\hspace*{4em} (z_1-z_2)^{l-1}(z_2-z_3)^{q-l}
   \{\{A(z_1),B(z_2)\},C(z_3)\}\nonu
&=&\sum_{l=1}^q \bin{q-1}{l-1} \{\{AB\}_l\ C\}_{p+q-l}(z_3)\}\,.
\label{eq:PBJacR}
\eea
These equations are of exactly the same form as the associativity conditions
for OPEs (see \re{eq:OPEJacRAB} with $q,p>0$).

An important difference with OPEs is that no ``regular" part is defined
for Poisson brackets. In particular, normal ordering is not necessary.
A Poisson bracket where a product of fields is involved, is simply\col
\be
\{A(z),B(w) C(w)\} = \{A(z),B(w)\} C(w) +
(-1)^{|A|\,|B|} B(w)\{A(z), C(w)\}\,.\label{eq:PBJacRreg}
\ee
When using the notation\col
\bea
\{AB\}_{-n}(z) = {1\over n!} (\del^nA(z))\, B(z)\,,&&n\geq 0\,,
\label{eq:PBregdef}
\eea
we see that \vgl{eq:PBJacRreg} corresponds to \vgl{eq:OPEJacRAB} for $p=0$
and with the double contractions $l<q$ dropped. This is also true for a
classical version of \vgl{eq:OPEJacL}. We can conclude that when using the
correspondence\col
\bea
\{AB\}_n \leftrightarrow [AB]_n\,,
\eea
computing with Poisson brackets follows almost the same
rules as used for OPEs\col\ one should drop double contractions and use
a graded-commutative and associative normal ordering. In
particular, as the Jaco\-bi--iden\-ti\-ties are the same, any
linear PB--al\-ge\-bra corresponds to an operator product algebra and
vice-versa. For nonlinear algebras, this is no longer true because of normal
ordering.

Due to this correspondence, we will often write ``classical OPEs'' for
Poisson brackets.

\section{Mode algebra\label{sct:modeAlgebra}}
In this section, we show that there is an infinite dimensional
algebra with a graded--symmetric bracket associated to every OPA.
For every operator $A$ we define the $m$-th mode of $A$ by
specifying how it acts on an operator\col
\be
\mo{A}{m} B \equiv [AB]_{m+a} \label{eq:modedef}\,,
\ee
where $a$ is the conformal dimension of $A$. The shift in the index is made
such that $\mo{A}{m} B$ has dimension $b-m$, independent of
$A$\ \footnote{This definition assumes that the OPA is graded, def.
\ref{def:dimension}. Of course, the mode algebra can also be defined without
this concept.}. Hence for operators with (half-)integer dimension, $m$ is
(half-)integer. An immediate consequence of this definition follows by
considering (\ref{eq:PolederL})\col
\be
\mo{(\partial A)}{m} = -(m+a) \mo{A}{m} \,.\label{eq:modeder}
\ee
We can now compute the graded commutator of two modes\col
\be
[\mo{A}{m},\mo{B}{n}] C
=[A[BC]_{n+b}]_{a+m} - (-1)^{|A|\,|B|} [B[AC]_{m+a}]_{n+b}\,.
\ee
Using the associativity condition \re{eq:OPEJacRAB} we see that\col
\be
[\mo{A}{m},\mo{B}{n}] = \sum_{l>0}\bin{m+a-1}{l-1}
\mo{({[AB]_l})}{m+n}\,. \label{eq:modecomm}
\ee
Note that this commutator is determined by the singular part of the OPE.
\begin{theorem}
Eq.\ \re{eq:modecomm} defines a graded commutator.
\end{theorem}
\begin{proof}
We can use the relation between the OPEs $AB$ and $BA$ \re{eq:OPEcomm} and
\vgl{eq:modeder} to show that\col
\bea
\leqn{-(-1)^{|A|\,|B|} [\mo{B}{n},\mo{A}{m}]}
&=&\sum_{l>0} \bin{n+b-1}{l-1} \sum_{p>l}{(-1)^p\over (p-l)!}
   \mo{(\partial^{p-l}\![AB]_p)}{m+n}\nonu
&=&\sum_{l>0}\sum_{p>l} \bin{n\!+\!b\!-\!1}{l-1}
   \bin{m\!+\!n\!+\!a\!+\!b\!-\!l\!-\!1}{p-l}\mo{([AB]_p)}{m+n}
\nonumber \,.
\eea
Using \vgl{eq:BinoIdrs}, we see that this is equal to \re{eq:modecomm}.
\end{proof}
A first example of a mode algebra is given by the modes of a Virasoro operator
$T$ with OPE \re{eq:VirasoroOPE}. For historic reasons, we denote
these modes with $L$. We find using \vgl{eq:modecomm}\col
\be
[\mo{L}{m},\mo{L}{n}] = (m-n)\mo{L}{m+n} + {c\over 2} \bin{m+1}{3} \delta_{m+n}\,,
\label{eq:VirasoroMode}
\ee
where we used that the modes of the unit operator (which is implicit in the
fourth order pole of \re{eq:VirasoroOPE}) are given by\col
\be
\mo{\unity}{m} = \delta_m\,.
\ee
The infinite dimensional Lie algebra with commutator \re{eq:VirasoroMode} is
called the Virasoro algebra\puzzle{ \cite{VirasoroAlgebra}}. We see that this
algebra is a central extension of the algebra of the classical generators of
conformal transformations \re{eq:ClassicalVir}. The modes corresponding to
the global conformal transformations $\mo{L}{1}, \mo{L}{0}, \mo{L}{-1}$
form a finite dimensional subalgebra where the central extension drops
out.

Similarly, the OPE of $T$ with a primary field
$\Phi$ with dimension $h$ \re{eq:PrimaryFieldOPE} gives the following
commutator\col \be
[\mo{L}{m},\mo{\Phi}{n}] = ((h\!-\!1)m-n)\mo{\Phi}{m+n}\,.
\label{eq:PrimaryFieldMode}
\ee

We have the following important theorem.
\begin{theorem}
For a given OPA, the commutator of the corresponding mode algebra
satisfies graded Jacobi identities modulo modes of null fields\col \be
[\mo{A}{k},[\mo{B}{l},\mo{C}{m}]] =
(-1)^{|A|\,|B|} [\mo{B}{l},[\mo{A}{k},\mo{C}{m}]] +
[[\mo{A}{k},\mo{B}{l}],\mo{C}{m}]]\,.\label{eq:ModeJac}
\ee
\end{theorem}
\begin{proof}
The \lhs\ of \vgl{eq:ModeJac} is by definition \re{eq:modecomm} equal to\col
\bea
\leqn{[\mo{A}{k},[\mo{B}{l},\mo{C}{m}]]\,=}
&&\sum_{q>0}\sum_{p>0}\bin{k+a-1}{q-1}\bin{l+b-1}{p-1}
    \mo{([A[BC]_p]_q)}{k+l+m}\,.
\eea
We can now use the associativity condition \re{eq:OPEJacRAB}.
Calling the summation index in \re{eq:OPEJacRAB} $r$ and renaming
$q=s+r-p$, it follows that
the Jacobi identity \re{eq:ModeJac} will be satisfied if\col
\bea
\leqn{\sum_{p>0}\bin{k+a-1}{r+s-p-1}\bin{l+b-1}{p-1}
\bin{r+s-p-1}{r-1}\,=} &&\bin{k+a-1}{r-1}\bin{k+l+a+b-r-1}{s-1}\,.
\eea
After cancelling out factors, one sees that this equation follows from
\vgl{eq:BinoIdnm}.
\end{proof}
Moreover, from the above proof it is clear that the reverse is also
true\col
\begin{theorem}
If the mode algebra satisfies the Jacobi-identities \re{eq:ModeJac} up to
null fields, the associativity conditions \re{eq:OPEJacRAB} are satisfied.
\end{theorem}
Modes of normal ordered operators are given by \vgl{eq:OPEJacL}\col
\be
\mo{([AB]_0)}{m} = \sum_l :\mo{A}{l} \mo{B}{m-l}:\,,\label{eq:modeNO}
\ee
where
\be
:\mo{A}{l} \mo{B}{m}:\ \  \equiv\,
   \left\{ \begin{array}{lcr}
    \mo{A}{l} \mo{B}{m}&&{\rm if }\,\,\,\,l\leq -a\\[3mm]
     (-1)^{|A|\,|B|} \mo{B}{m}\mo{A}{l} &&{\rm if }\,\,\,\,l> -a
   \end{array}\right. \label{eq:mode::def}
\ee
Consider an OPA where the generators have OPEs whose
singular part contains composite operators. In this case, the mode algebra is
only an infinite dimensional (super-)Lie algebra when those composites are
viewed as new elements of the algebra, such that the commutators close
linearly. Otherwise, the commutators close only in the enveloping algebra of
the modes of the generators.

As the mode algebra contains the same information as the OPEs, one can
always choose which one uses in a certain computation. For linear algebras
(where the OPEs close on a finite number of noncomposite operators) modes
are very convenient. However, for nonlinear algebras the infinite sums in
the modes of a composite are more difficult to handle.

The definition \vgl{eq:modedef} provides a realisation of the mode algebra.
In a canonical quantisation scheme, another representation is found in
terms of the creation- and annihilation operators. For different
periodicity conditions  in the coordinate $\sigma$ (relating $z$ and
$\exp(2i\pi)z$) of the symmetry generators, formally the same algebra arises.
However, the range of the indices differs. As an example, the $N=1$
superconformal algebra consist of a Virasoro operator $T$ and a fermionic
dimension $3/2$ primary operator $G$ with OPE\col
\be
G\times G\ =\ \opethree{{2c\over 3} \unity}{0}{2T}\label{eq:N=1OPE}\,.
\ee
This gives for the anticommutator of the modes\col
\be
[\mo{G}{m},\mo{G}{n}] = {2c\over 3}\bin{m+1/2}{2} \d_{m+n} + 2 \mo{L}{m+n}\,.
\ee
In the representation \vgl{eq:modedef} defined via the OPEs, $m$ and $n$
in this commutator are half-integer numbers. However, the algebra is also
well-defined if $m$ and $n$ are integer. This corresponds to different
boundary conditions on $G(z)$. The relation between the different modings
of the linear superconformal algebras is studied in 
\cite{SS:moding,moding}. 
We will use the notation $\mo{A}{m}$ in the representation
\vgl{eq:modedef} of the mode algebra, and drop the hats otherwise.

\section{Generating functionals \label{ssct:ZGamma}}
In this section, we will define the generating functionals for the
correlation functions of a conformal field theory and show that the Ward
identities give a set of functional equations for these functionals.

Consider a conformal field theory with fields $\phi_i$ and action
$S[\phi_i]$. We denote the generators of the chiral symmetries of
the theory with $T_k$. The partition function $Z$ is defined by\col
\bea
Z[\mu] &=& {1\over\cn}\int [d\varphi_i]\,
    \exp \left(-S[\varphi_j] - {1\over\pi}\int d^2 x \, \mu^k(x)
       T_k(x)\right) \label{eq:Zdef}\\
&=&  < \exp \left(-{1\over\pi}\int d^2 x \,
       \mu^k(x)T_k(x)\right)>_{\rm OPE}\, .
\eea
Here the normalisation constant $\cn$ was defined in \vgl{eq:normPathdef}
and $\mu^k$ (the ``sources") are non-fluctuating fields. $Z$ is the generating
functional for the correlation functions of the generators $T_k$. Indeed,
by functional derivation with respect to the sources we can determine
every correlation function\col
 \be
<T_1(x_1) T_2(x_2) T_3(x_3)> \,=\,
        (-\pi)^3 \,\dd{\mu^1(x_1)} \dd{\mu^2(x_2)} \dd{\mu^3(x_3)} Z[\mu]
\Bigr| \raisebox{-1.5ex}{$\mu=0$}\, .
\ee
If a generator $T_k$ is fermionic, $\mu^k$ is Gras\-mann-odd, \ie
anticommuting. We will always use left-functional derivatives in this work.

It is often useful to define the induced action $\Gamma$ as the generating
functional of all ``connected" diagrams\col
\be
Z[\mu] = \exp (-\Gamma[\mu])\label{eq:indActdef}\,.
\ee

We can view the sources $\mu^k$ as gauge fields. By assigning appropriate
transformation rules for $\mu^k$, we can try to make the global symmetries
local at the classical level. As an example, in a conformal field theory,
the action is invariant under the conformal transformations generated by
$T$, when the parameter $\e^z$ is analytic. If we want to make the
theory invariant for any $\e^z$, we have to couple the generator $T$ to a
gauge field. We find that\col
\be
 \delta_\e (S + {1\over\pi} \int\mu T) =
    {1\over\pi}\int\left(-\db \e T + \delta\mu T + \mu \delta T\right)\, ,
\ee
where we used the definition of $T$ as a Noether current
(\vgl{eq:defTgenTrans}). Applying the transformation rule for $T$
(\vgl{eq:TTransfInf}) in the classical case $(c=0)$, we see that if the
source $\mu$ transforms as\col
\be
\delta_\e \mu = \db \e +
\e \dz \mu - \dz \e \mu \, ,\label{eq:defTmutransf}
\ee
the action $S + {1\over\pi} \int\mu T$ is classically invariant. When
gauging not only the conformal symmetry, we would expect that
higher order terms in the sources have to be added to \vgl{eq:Zdef} to
obtain invariance. However, Hull \cite{Hull91} proved that minimal
coupling (addition of only linear terms) is sufficient to gauge chiral
symmetries.

It is possible that the resulting local symmetry does not survive at the
quantum level. For example, the Schwinger term in the transformation law of
$T$ (\vgl{eq:TTransfInf}) breaks gauge invariance. In general, central terms in
the transformation laws of the symmetry generators give rise to
 ``universal'' anomalies, which cannot be canceled by changing the
transformation law of the gauge fields\footnote{The universal anomalies
could be canceled by including a gauge field for the ``generator''
$\unity$.}. In this case, the induced action
$\Gamma$ is a (in general nonlocal) functional of the gauge fields $\mu^k$
and is used in $\Ww$--gravity theories (see chapter
\ref{chp:renormalisations}) and non-critical \WS s.

The Ward identities can be used to derive functional equations for the
generating functionals $Z$ and $\Gamma$. As an example, consider the
induced action where only the \emt\ is coupled to a source (\ie the
generating functional for correlation functions with only $T$).  Under
the variation of the source (\vgl{eq:defTmutransf}), the partition
function transforms as\col
\bea
\delta Z[\mu]&=& <\exp\left(
   -{1\over\pi}\int\left(\mu+\delta\mu\right)T\right)> -  Z[\mu]\nonu
      &=& -{1\over\pi} \int  (\db \e + \e \dz \mu -(\dz\e)\mu)
         <T \,\exp \left(-{1\over\pi}\int \mu T\right)>  \label{eq:deltaZT1}
\eea
To compute the first integral with $\db \e T$, we note that
in the complex basis, when $\e^\zb=0$ and all $\Phi_i=T$, \vgl{eq:WardIdTij}
becomes\col
\bea
\leqn{-{1\over \pi}\int d^2x\, \db\e(x)\, <T(x) T(x_1) \cdots T(x_N)>}
& =&
\sum_{j=1}^N <T(x_1)\cdots \delta_{\e}T(x_j) \cdots T(x_N)>\,.
\label{eq:dZT1nieuw}
\eea
We can now multiply this equation with $\mu(x_1)\cdots \mu(x_N)$ and
integrate over all $x_j$. Using crossing symmetry in the \rhs\ of
\vgl{eq:dZT1nieuw}, we get (in an obvious notation)\col
\bea
\leqn{ \int d^2x\,
      \db\e(x)\,<T(x) \left(-{1\over\pi}\int \mu T\right)^N>}
& =& N\int d^2x\, \mu(x)
      <\delta_{\e}T(x) \left(-{1\over\pi}\int \mu T\right)^{N-1}>\, .
\eea
This gives the following result\col
\bea
\leqn{\int d^2x\, \db\e(x)\, <T(x)\,
       \exp \left(-{1\over\pi}\int \mu T\right)>\, =}
 &&\int d^2x\, \mu(x) <\delta_{\e}T(x)  \exp
       \left(-{1\over\pi}\int \mu T\right)>\,.
\eea
Using \vgl{eq:TTransfInf}, the variation of the partition function
\re{eq:deltaZT1} becomes\col
\be
\delta Z[\mu] = {c\over 12\pi}\left(\int\, \e\dz^3\mu\right)
\,Z[\mu]\,. \label{eq:deltaZT}
\ee

On the other hand, we can rewrite the \rhs\ of \vgl{eq:deltaZT1} using\col
\be
<T(x) \,\exp \left(-{1\over\pi}\int \mu T\right)> =
 -\pi\ddt{Z}{\mu(x)}[\mu]=\pi Z[\mu] \ddt{\Gamma}{\mu(x)}[\mu]
\,. \label{eq:deltaZT2}
\ee
Combining \re{eq:deltaZT1} and \re{eq:deltaZT2} gives us a functional
equation for $Z$, and thus for $\Gamma$\col
\be
{c\over 12\pi}\dz^3 \mu = \db t - \mu \dz t - 2 (\dz \mu) t
\label{eq:WardIdTGamma}
\ee
where\col
\be
t = {\delta \Gamma  \over \delta \mu}\,.
\ee
We see that the Ward identities fix the generating functionals. This is of
course no surprise, as we already knew that the Ward identities determine the
correlation functions.  The underlying theory $S[\varphi_j]$ determines  the
central charge $c$.

By rescaling $t$ in \vgl{eq:WardIdTGamma}, we can make the functional
equation $c$-independent. We see that $\Gamma = c \Gamma^{(0)}$, where
$\Gamma^{(0)}$ is $c$-independent. In fact, the solution of the Ward identity
\re{eq:WardIdTGamma} was given by Polya\-kov \cite{PolyWardT}\col
\be
\Gamma[\mu] = {c\over 24\pi}\int
     \mu \dz^2\,(1-\db^{-1}\mu \dz)^{-1} {\dz\over\db} \mu \,
,\label{eq:explicitGammaT}
\ee
where the inverse derivative is defined in \vgl{eq:inversedb}. For a general
set of symmetry generators, it will not be possible to solve the functional
equation in closed form.

We now show how OPEs can be used to derive the functional equation on the
partition function $Z$ (\vgl{eq:Zdef}). We start by computing\col
\be
-\pi \db \ddt{Z}{\mu^i(z,\zb)} =
   \db < T_i(z) \exp \left(-{1\over\pi}\int d^2 x \, \mu^k(x)
T_k(x)\right)>\,.
\ee
Let us look at order $N$ in the sources\col
\bea
\leqn{\db < T_i(z)\left(-\int \mu^k T_k\right)^N>}
&=&-N \int d^2x_0\, \mu^j(x_0)
       \db <\sum_{n>0} {[T_iT_j]_{n}(z_0)\over (z-z_0)^{n} }
          \left(-\int\mu^k T_k\right)^{N-1}>\nonu
  &=& \pi N \sum_{n> 0} {(-1)^{n}  \over (n-1)!}\nonu
  &&\hspace*{2em} \dz^{n-1} \left(\mu^j(z,\zb)
        <[T_iT_j]_{n}(z)\left(-\int\mu^kT_k\right)^{N-1}\!\!\!>\right)\nonu
  &=& \pi N \sum_{n> 0} {\dz^{n-1}\mu^j(z,\zb)\over(n-1)!}
%  \nonu
%  &&\hspace*{2em}
       <[T_jT_i]_n(z)\left(-\int\mu^kT_k\right)^{N-1}\!\!\!>\,,
\eea
where we used \vgl{eq:GreenzGen} in the second step, and the last step
\cite{JdBthesis} relies on \vgl{eq:OPEcomm}. Hence,
\bea
%\leqn{
-\pi \db \ddt{Z}{\mu^i(z,\zb)} =
   \sum_{n> 0}\! {\left(\dz^{n-1}\mu^j(z,\zb)\right)\over(n-1)!}\!
%   \hspace*{3em}}
%&&\hspace*{3em}
       <\![T_jT_i]_n(z)\exp\left(-{1\over\pi}\int\mu^kT_k\right)\!\!\!>,
\label{eq:WIdOPE}
\eea
Note that only the singular parts of the OPEs contribute. One can
easily check that \vgl{eq:WIdOPE} reproduces the functional equation
\re{eq:WardIdTGamma}.

Consider now the case of a linear algebra, \ie the singular parts of the OPEs
contain only the generators $T_j$ or their derivatives. The result
\re{eq:WIdOPE} can then be expressed in terms of functional derivatives with
respect to $\mu^j$ of $Z$ as in \vgl{eq:deltaZT2}.
Central extension terms in \vgl{eq:WIdOPE} are simply proportional to $Z$.
This means an overall factor of $Z$ can be divided out. If all central
extension terms are proportional to $c$, we again infer that $\Gamma = c
\Gamma^{(0)}$.

When the OPEs close nonlinearly, \ie contain also normal ordered expressions
of the generators, it is still possible to write down a functional equation
for $Z$ or $\Gamma$ by using \vgl{eq:pointsplreg}. An explicit calculation
of such a case is given in section \ref{sct:FFexamples}. The resulting
functional equations have not been solved up to now. We will treat this problem
at several points in this work, but especially in chapter
\ref{chp:renormalisations}.



\section{A few examples\label{sct:FreeFields}}
\subsection{The free massless scalar\label{ssct:FreeScalar}}
The action for a massless scalar propagating in a space with metric
$g_{ij}$ is given by\col
\puzzle{JdB p.17 8pi}
\be
S_s[X,g^{ij}] = -{1\over 4\pi\lambda}\int dx^2\,\sqrt{g}
g^{ij}\partial_iX(x)\partial_jX(x)\,, \label{eq:actionFreeFieldg}
\ee
with $g$ the absolute value of the determinant of $g_{ij}$ and
$\lambda$ a normalisation constant. For $D$ scalars, this gives an action
for the bosonic string in $D$ dimensions with flat target space, see
chapter \ref{chp:W-strings}.

Before discussing the Ward identity of the \emt\ for this action, we
first observe that the action \re{eq:actionFreeFieldg} has a symmetry\col
%de Wit p.57
\be \delta X(x) = \e(x)\,.
\ee
We will follow the reasoning of subsection \ref{ssct:WardId} to find the
Ward identity corresponding to this symmetry. The
Noether current for this symmetry is ${1\over 2\lambda}\partial_iX(x)$ (see
\vgl{eq:defTgenTrans}).
It satisfies the Ward identity (see \vgl{eq:WardIdTij})\col
\bea
\leqn{-{1\over 2\pi\lambda}\int d^2 x \,\sqrt{g} g^{ij}\partial_i\e(x)
<\partial_jX(x) X(x_1)X(x_2)\cdots>=}
&&<\e(x_1)X(x_2)\cdots> + <X(x_1)\e(x_2)\cdots> + \cdots\label{eq:WardIddX}
\eea
This symmetry has a chiral component $\dz X$
for which the equation corresponding to \re{eq:WardIdTPhiPhiseries} is\col
\bea
\leqn{<\dz X(z,\zb) X(z_1,\zb_1)X(z_2,\zb_2)\cdots>=}
&&{\lambda\over z-z_1}<X(z_2,\zb_2)\cdots> +
  {\lambda\over z-z_2}<X(z_1,\zb_1)\cdots>+\cdots
\eea
corresponding to the OPE\footnote{As before, we use the misleading notation
$\dz X(z)$.}\col
\be \dz X(z) X(z_0,\zb_0) = {\lambda\over z-z_0} + O(z-z_0)^0\,.
\label{eq:dzXXOPE}
\ee
This agrees with the propagator for $X$ which is given by the
Green's function for the Laplacian (see appendix \ref{app:Green})\col
\be
<X(z,\zb) X(z_0,\zb_0)> = \lambda\log(z-z_0)(\zb-\zb_0)\,. \label{eq:XXOPE}
\ee
Note that $X$ itself is a field whose two-point function is not a
holomorphic function of $z$ or $\zb$. This in fact makes it impossible to
use the OPE formalism with the field $X$. This need not surprise us, as $X$
not a symmetry generator. We will only work with the chiral symmetry
generator $\dz X$. For easy reference we give its OPE, which follows from
\vgl{eq:dzXXOPE} by taking an additional derivative\col
\bea
 \dz X(z) \dz X(z_0) &=& {\lambda\over (z-z_0)^2} + O(z-z_0)^0
\label{eq:dzXdzXOPE}
\eea

The action \re{eq:actionFreeFieldg} is invariant under the transformation
$\delta_\e X = \e^i\partial_iX$. Hence, we have that the \emt\
\vgl{eq:defT} is classically conserved. It is given by\col
\be
T_{ij} = {1\over 2\lambda}\partial_iX(x)\partial_jX(x)\ -
 {1\over 4\lambda} g_{ij} g^{kl}  \partial_kX(x)\partial_lX(x)\,.
\ee
In the complex basis, $T_{ij}$ has only two non-vanishing components\col
\be
T_{zz} = {1\over 2\lambda}\dz X \dz X\,,\hspace*{5em}
T_{\zb\zb}={1\over 2\lambda}\db X \db X\,.\label{eq:TFreeScalar}
\ee
However, these expression are not well-defined in the quantum case, due to
short-distance singularities. We use point-splitting regularisation  to
define the quantum operator. In the OPE formalism, this becomes
$T={1\over 2\lambda}[\dz X \dz X]_0$. We can now use the OPE
\re{eq:dzXdzXOPE}, and the rules of subsection \ref{ssct:OPErules} to
compute the OPE of $T$ with itself. One ends up with the correct Virasoro
OPE \re{eq:VirasoroOPE}, with a central charge $c=1$. Furthermore, it is
easy to check that $X$ is a primary field with respect to $T$ of
dimension zero. Moreover, $\dz X$ is also primary, having dimension one.
In the rest of this subsection we set $\lambda=1$.

Let us now check for the mode algebra \vgl{eq:modecomm} corresponding to the
OPE of $\dz X$ with itself \vgl{eq:dzXdzXOPE}. The modes of $\dz X$ are
traditionally denoted with $\alpha_m$. We find\col \be
[\alpha_{m},\alpha_{n}]\,=\,m \delta_{m+n}\,,
\ee
which is related to the standard harmonic oscillator
commutation relations via a rescaling with $\sqrt{|m|}$.

In string theory, computing scattering amplitudes is done by inserting local
operators of the correct momentum in the path integral and integrating over
the coordinates \cite{Kaku,GSW}. These local operators are (composites with)
normal ordered exponentials of the scalar field. Because $X$ cannot be
treated in the present OPE scheme, we should resort to different techniques
to define these exponentials. This is most conveniently done using a mode
expansion of $X$. One shows that one can define a chiral operator which we
write symbolically as\col
\be V_a(z)\ =\ :\!\exp( a X(z))\!:\,,
\hspace*{3em} a\in\Bbb{C}\label{eq:vertexdef}\,,
\ee
for which the following identities hold\col
\bea
&&\dz V_a\ =\  a :\!\dz X\,V_a \!:\label{eq:vertexder} \\
&&\dz X(z)\, V_a(w)\ =\ a {V_a(w)\over (z-w)} + O(z-w)^0
\label{eq:dzXvertexOPE}
\eea
and
\bea
\leqn{V_a(z)\, V_b(w)}
&=& (z-w)^{a b} :\exp (a X(z))\ \exp(b X(w)):\nonu
&=&(z\!-\!w)^{a b}:\Big(1 +  (z\!-\!w) a\dz X(w) +.\nonu
&&\hspace*{2em}
    {(z\!-\!w)^2 \over 2}(a\dz^2 X(w)+a^2 \dz X(w) \dz X(w))
 \ldots\Big) V_{a+b}(w):\,, \label{eq:vertexOPE}
\eea
where the last line is a well-defined expression where normal ordering, as
defined in \vgl{eq:pointsplreg}, from right to left is understood.
\puzzle{??refs deWit free fields, other.} We will use these equations as the
definition for the vertex operators.

Using the eqs.\ \re{eq:vertexder} and \re{eq:dzXvertexOPE}, it is easy to
check that $V_a$ is primary with conformal dimension $a^2/2$ with respect to
$T$ \re{eq:TFreeScalar}. This differs from the classical dimension which is
zero as $X$ has dimension zero.\puzzle{show that k is the momentum}

The OPE \re{eq:vertexOPE} is distinctly different from the
ones considered before. Indeed, noninteger poles are possible. The
rules constructed in section \ref{ssct:OPErules} are not valid for such
a case. However, every OPE with vertex operators we will need, will
follow immediately from the above definitions. Another peculiarity is that
when $ab \in \Bbb{Z}$, the definition \re{eq:vertexOPE} fixes the regular
part of the OPE of two vertex operators in terms of normal ordered products
of $\dz X$ and a vertex operator.
An interesting example of this is for $V_{\pm 1}$\col
\bea
V_{\pm 1}(z)\,V_{\pm 1}(w) &=& (z-w) V_2(w) + \ldots\nonu
V_{\pm 1}(z)\,V_{\mp 1}(w) &=& {1\over (z-w)}\, \pm\, \dz X(w) +
\ldots\label{eq:fermionisation1} \eea
We see that we have two operators\col
\bea
{V_1 + V_{-1}\over \sqrt{2}}=\sqrt{2}\cosh X&{\rm and}&
{V_1 - V_{-1}\over \sqrt{2}i}=-i\sqrt{2}\sinh X
\label{eq:bosonisation}
\eea
which satisfy the OPEs of two free fermions (see \vgl {eq:psipsiOPE}
below). \puzzle{ ??refs. } Vertex operators can also be used to
construct realisations of affine Lie algebras, as
we will see in section \ref{sct:WB2}.

When considering realisations of \WA s using scalars, it is obviously
a problem that the \emt\ of the free scalars is restricted to an integer
central charge $n$ by considering $n$ free bosons. However, we can modify
the \vgl{eq:TFreeScalar} to\col \be
T ={1\over 2}:\dz X \dz X: - q \dz^2 X\,,\label{eq:Tbackgroundcharge}
\ee
which has a central charge $1-12q^2$. This corresponds to adding a
background charge to the free field
action \cite{FeiFubackch,FMS,DotFatbackch}.

\subsection{The free Majorana fermion\label{ssct:FreeFermion}}
The free Majorana fermion in two dimensions has the following action in
the conformal gauge\col
\puzzle{$\sqrt{g} ?$}
\be
S_f[\psi]={1\over 2\pi\lambda}\int d^2x\,\psi(x)\db\psi(x)\,,
\label{eq:actionFreeFermion}
\ee
where $\lambda$ is a normalisation constant. The equation of motion for
$\psi$ is\col \be
\db \psi=0\,.
\ee
Hence, $\psi$ is a chiral field and we will write $\psi(z)$. Note that
under conformal transformations it has to transform as a primary field of
dimension $h=1/2$, $\hb=0$ to have a conformal invariant action.
$-{1\over\lambda}\psi(x)$ is the Noether current corresponding to the
transformation\col
\be
\d_\e\psi(x) = \e(x)\,,
\ee
where $\e$ is a Grasmann--odd field.
Although we can proceed as before, we will derive the OPEs of $\psi$ using
a different technique. Consider the generating functional (see
\vgl{eq:Zdef})\col
\be
 Z[\mu] = {1\over\cn}\int [d\psi]\,
    \exp \left(-S_f[\psi] - {1\over\pi}\int d^2 x \, \mu(x) \psi(x)\right)
\,,
\ee
where $\mu$ is Grasmann--odd. Such a pathintegral can be computed
by converting it to a Gaussian path integral, as we will now show. We start
by rewriting the exponent in the following way\col
\bea
Z[\mu]&=&  {1\over\cn}\int [d\psi]\,
    \exp \left(-{1\over 2\pi\lambda}\int
       (\psi+\lambda(\db^{-1}\mu))\db
          (\psi+\lambda\db^{-1}\mu)\right)\nonu
&&    \exp \left({\lambda \over 2\pi}\int (\db^{-1}\mu) \mu\right)\,,
\eea
where we used the definition of the inverse derivative of appendix
\ref{app:Green}, supposing that $\mu$ decays sufficiently fast at infinity.
One can now shift the integration variables to
$\tilde{\psi}=\psi+\lambda\db^{-1}\mu$. This shift has a Jacobian $1$.
One arrives at\col
\be
Z[\mu]=  {1\over\cn}\int [d\tilde{\psi}]\,
    \exp\left( {-1\over 2\lambda\pi}\int
      \tilde{\psi}\db \tilde{\psi}\right)
   \exp \left({\lambda\over 2\pi}\int (\db^{-1}\mu) \mu\right)\,.
\ee
The remaining path integral cancels $\cn$ (see \vgl{eq:normPathdef}) and
we have\col \be
Z[\mu] =\exp \left(-{\lambda\over 2\pi^2}
        \int dx^2 dx_0^2\ \mu(z,\zb) {1\over z-z_0} \mu(z_0,\zb_0) \right)\,.
\label{eq:ZFreeFermion}
\ee
From this result we immediately see that the one-point function $<\psi(x)>$
is zero, while the two-point function is given by\col
\bea
<\psi(z) \psi(z_0)> &=&
\pi^2 \dd{\mu(z,\zb)} \dd{\mu(z_0,\zb_0)}
  Z[\mu]\Bigr| \raisebox{-1.5ex}{$\mu=0$}\nonu
&=& {\lambda \over z-z_0}\,,
\eea
where we used left-functional derivatives. The corresponding OPE
is\footnote{It is easy to prove that this OPE gives rise to a functional
equation for $Z[\mu]$ as in subsection \ref{ssct:ZGamma}, which is satisfied by
\vgl{eq:ZFreeFermion}.}\col
\be \psi(z)\,\psi(w)={\lambda\over z-w}+O(z-w)^0\,.\label{eq:psipsiOPE}
\ee
To find the \emt\ for the action \re{eq:actionFreeFermion}, we use the
definition \re{eq:defTgenTrans} with $\e^i$ decaying at
infinity, together with the transformation law \re{eq:PrimaryTransfInf} for
$\psi$. We find\col \bea
\delta S_f&=&{1\over \lambda \pi}\int d^2x\,\left(\e^z \dz \psi +
      1/2 \dz \e^z \psi + \e^{\zb} \db \psi\right)\db\psi\nonu
&=&{1\over 2\lambda \pi}\int d^2x\, \db\e^z \psi\dz\psi\,.
\eea
Hence, we find only one component $T^{\zb\zb}$ non-zero,
according to \re{eq:TTbdef} we have\col \be
 T\ =\ {1\over 2\lambda}:\dz\psi\,\psi:\,,\label{eq:TFreeFermion} \ee
which has central charge $c=1/2$.

\subsection{Other first order systems\label{ssct:bc}}
We will need two other first order systems, the fermionic $b,c$ and the
bosonic $\beta,\gamma$ system \cite{FMS}. They will arise as the ghosts in
the BRST-quantisation of systems with conformal invariance. We will combine
both using a supermatrix notation (see appendix \ref{app:super}). The action
is\col \be
S_{\rm bc}[b,c]={1\over \pi}\int d^2x\,b^i(x) \,{}_i A^j\,\db({}_jc)(x)\,,
\label{eq:actionbc} \ee
where ${}_i A^j$ is a constant bosonic supermatrix which is invertible. We
assume that it does not mix fermions and bosons. We also take $b$ and
$c$ to be bosonic matrices\footnote{One can of course take $b$ and $c$ to
be fermionic matrices. The two-point function \re{eq:OPEbc} does not
depend on this convention.}. The equations of motion again show that
$b^i,c_i$ are chiral fields. Conformal invariance requires the $b^i$ to be
primary fields with dimension $h^i$ and $c_i$ also to be primary such that
${\rm dim}({}_i A^j\ {}_jc)=1-h^i$ (and zero for $\hb$). We proceed now as
in the previous subsection\col  \bea
 Z[\mu,\nu] &=& {1\over\cn}\int [db^i][dc_j]\,
    \exp -\left(S_{\rm bc}[b,c] + {1\over\pi}\int \mu_k \,{}^kb + \nu^k\,{}_kc
\right) \nonu
&=&  {1\over\cn}\int [db^i][dc_j]\,
    \exp \left({1\over \pi}\int
       (b+\db^{-1}\nu A^{-1})A\db
          (c-(A\db)^{-1}\mu))\right)\nonu
&&   \qquad \exp \left({1 \over \pi}\int (\db^{-1}\nu) A^{-1}\mu\right)\nonu
&=&\exp \left(-{1 \over \pi}\int d^2x d^2x_0\
     \nu^i(z,\zb){{}_i (A^{-1})^j\over z-z_0}\ {}_j\mu(z_0,\zb_0)\right)\,.
\eea
Carefully keeping track of the signs, we get for the two-point functions\col
\bea
<{}_ic(z) \ b^j(z_0)> &=&
\pi^2 \dd{\nu^i(z,\zb)} \dd{\mu_j(z_0,\zb_0)}
Z[\mu,\nu]\Bigr|\raisebox{-1.5ex}{$\mu,\nu=0$}\nonu
 &=& {-{}_i(A^{-1})^j
\over z-z_0}\,.\label{eq:OPEbc} \eea
This OPE does not depend on the dimensions of the fields, while the
\emt\ of course does. To find the \emt, we proceed as in the previous
subsection. From\col
\be
\delta_\e \left( {}_i(Ac)\right)=\e^z \dz ({}_i(Ac)) +
(1-h^i)\dz\e^z\ {}_i(Ac)\,,
\ee
we get\col \be
T_{\rm bc}\,=\,: b A \dz c - (1-h^i)\dz(b^i\ {}_iA^j\ {}_jc):,\label{eq:Tbc}
\ee with central charge\col
\be
c=\sum_i (-1)^i 2(6(h^i)^2 - 6 h^i + 1))\,,\label{eq:ccbc}
\ee
where the phase factor is $-1$ for $b^i$ fermionic. Note that the second
term of \vgl{eq:Tbc} is the derivative of the ghost current. It is not
present in \vgl{eq:TFreeFermion} because $:\psi\psi:=0$.

\subsection{Wess-Zumino-Novikov-Witten models\label{sct:WZW}}

A final example of a conformal field theory is the \WZW\  \cite{WZW1,WZW2}.
It is a nonlinear sigma model with as target space a group manifold. The
(super) Lie group $\cg$ is required to be semisimple (see
\cite{Witten:WZWns,JMF:WZWns} for a weakening of this condition). We denote
its (super) Lie algebra with $\bar{g}$, see appendix \ref{app:super} for
conventions.

The WZWN action $\kappa S^+[g]$ is a functional of a $\cg$ -valued field
$g$ and is given by\col
\bea
\leqn{\kappa S^+ [g]=
\frac{\kappa}{4\pi x} \int_{\del \Omega} d^2 x \;
\str \left\{ \dz g^{-1} \db g \right\}}
&&+ \frac{\kappa}{12\pi x} \int_\Omega d^3 x\; \e^{\alpha\beta\gamma} \,
\str \left\{ g_{,\alpha} g^{-1} g_{,\beta}
g^{-1} g_{,\gamma} g^{-1} \right\}\,,
\label{eq:WZWs+def}
\eea
where $\Omega$ is a three-manifold with boundary $\del \Omega$.
It satisfies the Polyakov-Wiegman identity \cite{polwi}\col
\be
S^+[hg]=S^+[h]+S^+[g]-
   \frac{1}{2\pi x}\int \str
      \left( h^{-1}\dz h \db g g^{-1} \right)\,,\label{eq:RNpwfor}
\ee
which is obtained through direct computation.
We also introduce a functional $S^-[g]$ which is defined by\col
\be
S^-[g]=S^+[g^{-1}].
\ee
It is with this functional that we now continue. Using the
Polyakov-Wiegman identity \re{eq:RNpwfor}, we can show that the action
$S^-[g]$  transforms under\col
\be
\d_\eta g = \eta g\label{eq:detag}
\ee
as
\be
\d_\eta S^-[g] = -{\k \over 2\pi x}\int \str\left(
        \left(\dz g\, g^{-1}\right)\db \eta\right)\,.
\ee
We see that $S^-[g]$ is invariant when $\db\eta=0$. The corresponding
conserved current is\col
\be
J_z = {\k\over 2} \dz g\, g^{-1}\label{eq:WZWJgroupclassical}\,,
\ee
which is chiral, $\db J_z=0$.
Similarly, for $\d_{\bar{\eta}} g = g \bar{\eta}$ with $\dz\bar{\eta}=0$, we
find a conserved current\col
\be
J_\zb = -{\k\over 2}  g^{-1}\db g\label{eq:Jzbdef}\,.
\ee
$J_z$ transforms under \vgl{eq:detag} as\col
\bea
\d_\eta J_z &=& {\k\over 2} \dz \eta +
[\eta,J_z]\,,\label{eq:WZWdeltaJ}\\[2mm]
  &=& -\int dy\Big\{\str \left(\eta(y) J_z(y)\right), J_z(x)\Big\}_{\rm PB}
  \label{eq:WZWPB}
\eea
where the Poisson bracket is given by\footnote{We use the ``classical OPE''
notation by virtue of the correspondence discussed in subsection
\ref{ssct:Poisson}.}\col
\bea
J_z^a\times J_z^b& =& \opetwo{-{\k\over 2} g^{ab}}{
    {}^a g^d J_z^c\ {}_{cd}f^b}\,,\label{eq:KA}
\eea
which defines a current algebra of level $\k$.  It can be argued that the
relation \vgl{eq:KA} does not renormalise when going to the quantum
theory, and we will take \re{eq:KA} as the definition of the OPE.
However, the relation \re{eq:WZWJgroupclassical} can be renormalised to\col
\be
J_z=\frac {\alpha_\kappa} {2} \dz g\, g^{-1}\,.\label{eq:WZWJgroupquantum}
\ee
Using OPE techniques, it is argued in \cite{kz} that for a
current algebra of level $\kappa$, normalising the currents as in \re{eq:KA}
gives\col
\be
\alpha_\kappa =  \kappa+\tilde{h}\,,\label{eq:WZWkzadef}
\ee
where the dual Coxeter number $\tilde{h}$ is the eigenvalue of the quadratic
Casimir in the adjoint representation (see also appendix \ref{app:super}).
This follows from consistency requirements in the OPA of
the currents with $g$. We will need this renormalisation in chapter
\ref{chp:renormalisations}.

The OPA generated by the currents $J_z^a$ with the OPEs
\re{eq:KA} is known as a \KA\ or affine Lie algebra.
\KA s were studied in the mathematical literature \cite{KacB,Moody} much
earlier than \WZW s. For a review on the algebraic aspects of \KA s, see
\cite{GoddardOlive}. We will denote the \KA\ corresponding to $\bar{g}$ as
$\hat{g}$.

One can check that the Sugawara tensor\col
\be
T_S=\frac{1}{x\left(\kappa+\tilde{h} \right)}\str [J_zJ_z]_0\,,
\label{eq:Sugawara}
\ee
satisfies the Virasoro algebra with the central extension given by\col
\be
c=\frac{k(d_B-d_F)}{k+\tilde{h}}.
\ee
The currents $J^a_z$ have conformal dimension $1$ with respect to $T_S$.

We now briefly review some basic formulas for gauged \WZW s
\cite{WZW1,WZW2,polwi,orlando,dive} which we will need in chapter
\ref{chp:WZNW}. We can gauge the symmetries generated by $J_z$ by
introducing gauge fields $A_\zb$. The action\col
\be
\k S^-[g] + \frac{1}{\pi x} \int d^2 x \;
  \str \Big( J_z(x) A_{\zb}(x) \Big)
\ee
is (classically) invariant when the gauge fields transform as\col
\be
\d_\eta A_\zb = \db \eta + [\eta,A_\zb]\,.
\ee
The induced action
\re{eq:indActdef} $\Gamma[A_{\zb}]$ for the gauge fields $A_{\zb}$ is\col
\be
\exp\left( -\Gamma[A_{\zb}]\right) = \, \langle
  \exp \, - \frac{1}{\pi x} \int d^2 x \;
  \str \Big( J_z(x) A_{\zb}(x) \Big) \rangle \,.
\label{eq:RNtwelve}
\ee
By using the OPEs \vgl{eq:KA}, and the general formula for the functional
equation for $\Gamma$ \re{eq:WIdOPE} we find the following Ward identity\col
\be
\db  u_z - [A_{\zb}, u_z ] =\dz A_{\zb}\,,
\label{eq:WZWWardId}
\ee
where\col\puzzle{check}
\be
u^a_z(x)= - \frac{2\pi}{\kappa}g^{ab} \frac{\delta \Gamma[A_{\zb}]} {\delta
\,{}^b\! A_{\zb}(x)}\,. \label{eq:WZWudef}
\ee
The Ward identity is independent of $\kappa$, therefore\col
\be
\Gamma [A_{\zb}]=\kappa\Gamma^{(0)}[A_{\zb}],
\ee
where $\Gamma^{(0)}[A_{\zb}]$ is independent of $\kappa$. In
\cite{polwi,orlando}, it was observed that \vgl{eq:WZWWardId} states
that the curvature for the Yang-Mills fields $\{A,u\}$ vanishes.
% This
%condition is solved by parametrising $A_{\zb}$ as $A_{\zb}\equiv
%\db g g^{-1}$ and $u_z$ as $u_z\equiv\dz g g^{-1}$. Introducing
%the WZWN functional $S^+[g]$, we easily find that $\Gamma^{(0)} [A_{\zb}]$
%is given by\col
%\be
%\Gamma^{(0)} [A_{\zb}=\db g g^{-1}]=-S^+[g].
%\ee

%We could as well have performed the previous analysis starting from an
%anti-holomorphic affine Lie algebra. The induced action is then given by\col
%\be
%\exp\left( -\overline{\Gamma}[A_z]\right) = \, \langle
%  \exp \, - \frac{1}{\pi x} \int d^2 x \;
%  \str \left\{ J_{\zb}(x) A_z(x) \right\} \rangle ,
%\label{eq:RNatwelve}
%\ee
%where\col
%\be
%\overline{\Gamma}[A_z=\dz g g^{-1}]= -\kappa S^+[g^{-1}] \equiv -\kappa
%S^-[g]. \ee


\sectionappendix
\section{Appendix : Combinatorics \label{app:Combinatorics}}
This appendix combines some formulas for binomial coefficients and related
definitions. We define first the Pochhammer symbol\col
\be
(a)_n ={\Gamma(a+n)\over \Gamma(a)} = \prod_{j=0}^{n-1} (a+j)\,,\ \ \ \
a\in\Bbb{R},\,n\in\Bbb{N}
\label{eq:Pochdef}
\ee
where the last definition is also valid if $a$ is a negative integer. We
then define the binomial as\col
\bea
\bin{a}{n} = {(a)_n\over n!}&&a\in\Bbb{R}\,,\ n\in\Bbb{N}\,.\label{eq:Binodef}
\eea
In particular,
\be
\bin{-n}{m}=(-1)^m\bin{m+n-1}{m}\,,\ \ \ \ \ m,n\in\Bbb{N}\,.
\label{Binoneg}
\ee

We now list some identities for sums of binomial coefficients which are used
in section \ref{sct:modeAlgebra}.
 \be
\sum_{l\in\Bbb{N}} (-1)^l \bin{r}{l}\bin{r+s-l}{p-l} =
  \bin{s}{p}\,,\,\,\,\,\,r,s\in\Bbb{Z},\,p\in\Bbb{N}
\label{eq:BinoIdrs}
\ee
which can be proven by multiplying with $x^p$ and summing over $p$.
\be
\sum_{p\in\Bbb{N}}\bin{n}{p}\bin{m}{s-p} =
\bin{m+n}{s}\,,\,\,\,\,\,\,m,n\in\Bbb{Z},\,s\in\Bbb{N}
\label{eq:BinoIdnm}
\ee
which is proven by multiplying with $x^s$ and summing over $s$.

\sectionnormal
%12345678901234567890123456789012345678901234567890123456789012345678901234567890
\chapter{$\Ww$-algebras \label{chp:conformalInvariance}}
\mychapter
\def\ket{|\phi\!\!>}
\def\bra{<\!\!\phi|}
In this chapter, we focus our attention on \WA s.  Recall that we defined
a \WA\ as an Operator Product Algebra (OPA) generated by a Virasoro
operator $T$ and quasiprimary operators $W^i$. We first discuss the
representations of \WA s, section \ref{sct:WArepresentations}.
We then show how the representations of the global conformal group
restrict the possible Operator Product Expansions (OPEs) of the generators,
section \ref{sct:globalConformalRestrictions}. In the next section, the full
conformal group is studied. We demonstrate by using the Jacobi identities
how to reconstruct the complete OPEs from the knowledge of the
coefficients of the primary operators in these OPEs. The ultimate goal of
section \ref{sct:globalConformalRestrictions} is to provide the formulas
for an algorithmic computation of Virasoro descendants and the
coefficients with which they appear in OPEs.

In section \ref{sct:WAclassification}, we discuss
a few of the more important methods to construct \WA s and comment on the
classification of the \WA s. As an illustration of the ideas in this
chapter, we conclude with an example in section \ref{sct:WB2}\col\ the
$\Ww_c\!B_2$ algebra.

The presentation used in sections \ref{sct:globalConformalRestrictions}
and \ref{sct:localConformalRestrictions} is slightly more general than
the literature, and in subsections \ref{ssct:QPinOPE}, \ref{ssct:VirQP}
and \ref{ssct:descendantsOne} new developments are given. The results
on the $\Ww_c\!B_2$ algebra were published in \cite{wb2}.

\section{Introduction}
The chiral symmetry generators of a conformal field theory in general
form a \WA. The representation theory of \WA s will thus have its direct
consequences for the correlation functions of the theory. For physical
applications, highest weight representations are most useful.
In certain special
cases, the correlation functions are completely fixed by the symmetry
algebra. These representations are the minimal models, pioneered for the
Virasoro algebra in \cite{BPZ} and extended to other \WA s in
\cite{fazam:W3freefield,faly:WAn,lufa:HR,bbss3}. They consist of a
finite number of highest weight operators which each give rise to
degenerate a representation. Minimal models were first studied as
a simple model for infinite dimensional highest weight representations.
Almost all exactly solvable conformal field theories (CFT) are based on
minimal models. They also appear in $\Ww$-string theory, see chapter
\ref{chp:W-strings}.

The study of the \WA s is complicated by the fact that the OPEs can contain
composite terms, or equivalently the mode algebra is
nonlinear. The prototype of a nonlinear \WA\ is the $\Ww_3$ algebra
\cite{W3zamo}. It is generated by the \emt\ $T$ and a dimension $3$ current
$W$ with OPEs given by\col
\bea
T(z) T(w) &=& \frac{c}{2} (z-w)^{-4} + 2(z-w)^{-2} T(w)
    + (z-w)^{-1} \partial T(w) + \cdots
\nonu
T(z) W(w) &=& 3 (z-w)^{-2} W(w) + (z-w)^{-1} \partial W(w) +\cdots
\nonu
W(z) W(w) &=& \frac{c}{3} (z-w)^{-6}
    + 2 (z-w)^{-4} T(w) + (z-w)^{-3} \partial T(w)
\nonu
&& + (z-w)^{-2} \left[ 2 \beta \Lambda (w) + \frac{3}{10} \partial^2 T(w) \right]
\nonu
&& + (z-w)^{-1} \left[ \beta \partial \Lambda (w) + \frac{1}{15} \partial^3 T(w) \right]
   + \cdots ,
\label{eq:W3}
\eea
where\col
\be
\Lambda  = [TT]_0  - \frac{3}{10} \partial^2 T
\label{eq:Lambdadef}
\ee
and\col
\be
\beta = \frac{16}{22+5c}\,.
\label{eq:W3beta}
\ee
The operator $\Lambda$ is defined such that it is quasiprimary, \ie
$[T\Lambda]_3=0$.  Although this is the simplest nonlinear \WA\ for generic
$c$, checking the Jacobi identities by hand is already a nontrivial task.

We notice that the \rhs\ of the OPE $\FULL{WW}$ can be written in terms of
quasiprimaries or their derivatives.  Using the global conformal
transformations, generated by $L_{\pm 1},L_0$, we will see in section
\ref{sct:globalConformalRestrictions} that the coefficients of the
derivatives of the quasiprimaries, are numbers depending only on the dimensions of the operators
involved.  This result was already contained in the paper of Belavin,
Polyakov and Zamolodchikov \cite{BPZ}.  We present some new results on
how the quasiprimaries in a given OPE can be found.

The full conformal group gives even more information.  In case
that the OPA is generated by primary operators and their descendants,
the Jacobi identities with $\{T, \Psi_i, \Psi_j\}$ fix the form of the OPE
of two primaries $\Psi_i, \Psi_j$. In fact, the coefficients of the
descendants of the primaries occuring in the OPE now depend on the
dimensions of the operators and the central charge. In the example of
$\Ww_3$, the only primary in the singular part of the $W(z)W(w)$ OPE is
the unit operator at the sixth order pole. The rest of this OPE is then
completely determined by the Jacobi identity $\{T, W, W\}$. \\
It was shown in \cite{BPZ,josestany:W26} how the coefficients of the
Virasoro descendants can be found. As they can be very complicated,
automation of the computation of these coefficients is highly
desirable\footnote{The descendants themselves are easily computed using
\OPEdefs.}. However, in the basis for the descendants used in 
\cite{BPZ,josestany:W26} this computation is very CPU-intensive. 
We will therefore
construct a basis of quasiprimaries, and give the necessary
formulas to compute descendants and coefficients in this basis. A
\Mathematica\ package that implements these algorithms is now in testing
phase \cite{OPEconf}.

The conclusion will be
that we can reconstruct the OPEs of the primary
generators of a \WA\ from the list of the coefficients of all
primaries. For primary operators $\Psi_i$, we will write symbolically\col
\be
\Psi_i\times\Psi_j\longrightarrow C_{ij}{}^k\ [\Psi_k]\,,
\label{eq:OPEprimaries}
\ee
where $[\Psi_k]$ denotes the conformal family of the primary operator $\Psi_k$.
We only have to keep track of the primaries in the singular part of the
OPE, because any primaries in the regular part can be constructed from the
information contained in the singular part of the OPE. The structure
constants $C_{ij}{}^k$ are still restricted by the Jacobi identities
discussed in \ref{sct:OPEs}, but now for triples of primaries.
Unfortunately, the equations for the coefficients are too complicated to
solve in the general case, preventing to classify the \WA s in this way.
We will discuss briefly some other attempts towards the classification.

\section{Highest weight representations and minimal models
\label{sct:WArepresentations}}
In this section we consider representations of a \WA\ which are
constructed by acting repeatedly with the generators of the \WA\ on
a highest weight state $\ket$.
\begin{definition}\label{def:HWS}
For a mode algebra with generators $\MO{W^i}{n}$ (with $W^0\equiv T$),
a highest weight state (HWS) $\ket$ with weights $w^i$ satisfies\col
\[ \begin{array}{llll}
      \MO{W^i}{0}\ket\ =\ w^i\ket,&&
      \MO{W^i}{n}\ket\ =\ 0& n>0\,,
   \end{array} \]
\end{definition}
In this definition it is understood that a generator with half-integral
modes has no weight associated to it.
We will use the correspondence between states and fields in
meromorphic conformal field theory, as discussed in \cite{Goddard}. So, with
every highest weight operator $\phi$ corresponds a highest weight state
$\ket$.

For the Virasoro algebra it is natural to take $\phi$ to be a
primary operator of dimension $h$. This gives an extra
condition $\mo{L}{-1}\phi=\dz\phi$. The notion of a primary operator, and
in particular the requirement that the first order pole of the OPE of the
\emt\ with a primary operator is the derivative of the operator, arises
from the geometrical interpretation of a conformal transformation
\vgl{eq:ConfTransf}. As no geometrical meaning is currently known for the
transformations generated by currents with a nonlinear OPE, \puzzle{(for
some attempts see ??refs)} the concept of a $\Ww$--primary cannot be
defined at present. \puzzle{primary basis }

To define a highest weight representation, we first introduce some
notation. We will denote a sequence $\MO{(W^i)}{-n_1}\ldots
\MO{(W^i)}{-n_k}$ where $n_j\geq n_{j+1}>0$, as $\MO{(W^i)}{-\{n\}}$. This
notation is extended to sequences of modes of different generators
$\MO{W^0}{-\{n_0\}}\ldots \MO{W^d}{-\{n_d\}}$, which we write as
$\MO{W}{-\{n\}}$, where $\{n\}$ forms an ordered partition of $N$ with a
different ``colour" for every generator. We use the convention that
for a sequence of positive modes, the order is the reverse as for
negative modes, \ie $\MO{W}{\{n\}}$ acts like $\MO{W^d}{\{n^d_{k_d}\}}
\ldots \MO{W^d}{\{n^d_{1}\}}\ldots \MO{W^0}{\{n^0_{1}\}}$.

A Verma module is then defined as the space of all ``descendants''
$\MO{W}{-\{n\}}\ket$. The level of the descendant
is defined as its $L_0$ weight minus the $L_0$ weight of $\ket$,
\eg the level of $L_{-n} \MO{(W^i)}{-m} \phi$ is $n+m$.
Although the dimension of the Verma modula
is infinite, the dimension of the space of descendants at a
certain level is finite.

The representation can then be computed by using the (graded) commutators of
the modes and the definition \ref{def:HWS}.
The representation will (at least) depend on the weights $w^i$ and on the
central charge $c$ of the \WA, we will write $\car(w^i,c)$.


The representation is reducible, or degenerate, if at a certain level $N$
there occurs a new HWS, which will be the starting point for a new tower
of states. One can then divide out this new representation
$\car(\tilde{w}^i, c)$, where $\tilde{h}=\tilde{w}^0 =h+N$. If this
process is repeated for all descendant HWSs, one finally obtains an
irreducible representation. The descendant state which is also a HWS is
often called a singular vector of the representation.

We can define an inner product on the Verma module indirectly via
the adjoint operation\col
\bea
\left(\MO{W^i}{n}\right)^\dagger = \MO{W^i}{-n}
&&
<\!\!\phi\ket=1\,.\label{eq:modeadjointdef}
\eea
For two states with weights $w^i,\tilde{w}^i$, \vgl{eq:modeadjointdef} implies
that a nonzero inner product can only occur of $w^i=\tilde{w}^i$. In
particular, two descendant states of different level have zero inner product.
Singular vectors have inner product with all other states.

Singular vectors can be constructed using screening operators $S$. These are
characterised by the fact that the singular part of the OPE of a
generator $W^i$ with $S$ can be written as a total derivative\col
\be
W^i(z)S(w) = \sum_n \frac{[W^iS]_n(w)}{(z-w)^n} = \frac{d}{dw} [
\mbox{something}] + O(z-w)^0\, .\label{eq:screeningdef}
\ee
Using \vgl{eq:OPEcomm}, this is seen to be equivalent to $[SW^i]_1 =0$.
For any screening current $S$, we can define an
``intertwiner" $Q_S$ whose action on an operator $X$ is given by\col
\be
Q_S \, X(w)\ =\ \oint_{{\cal C}_w} {dz\over 2\pi i} S(z)\,X(w)\ = [SX]_1(w)\,.
\ee
Due to \vgl{eq:screeningdef}, $Q_S$ commutes with the modes $\MO{W^i}{m}$.
This means that $Q_S$ on a HWS $\ket$ is either zero, or another HWS. In
general, $Q_S\ket$ will be a descendant of another HWS. Hence, the
intertwiners can be used to construct singular vectors.

The matrix $S$ of the inproducts of all descendants at level $N$ is
called the \v{S}apovalov form. It depends only on the weights $w^i$ of the
HWS and the central charge of the \WA. The determinant of $S$ is the
Ka\v{c}--determinant \cite{KacB}. \puzzle{Due to theorem \ref{trm:HWSdesc}, }
Its zeroes are related to the singular vectors in the representation.

Because highest weight descendants are null operators, they generate partial
differential equations on the correlation functions \cite{BPZ}. Completely
degenerate representations have as much independent null vectors as
possible \cite{WAreviewpbks}. It can then be proven \cite{BPZ} that the
HWSs which give rise to a singular vector form a closed (possibly
non-meromorphic) OPA. Minimal models are then defined as those cases where
this OPA is finitely generated.

%algebra. As an example, the $W\!A_n$ algebras contain  besides the
%energy-momentum tensor, $n-1$ primary operators of dimension
%$3,4,\ldots,n+1$.  The central charge of the corresponding unitary
%minimal models is given by following discrete series \cite{faly:WAn}
%\begin{equation} c_{A_n} = n\left( 1-\frac{(n+1)(n+2)}{p(p+1)}\right),
%\end{equation}
%where $p=n+2,n+3,\ldots$.

\section{Consequences of the global conformal group
\label{sct:globalConformalRestrictions}}
In this section, we will study the OPEs of the quasiprimaries in a \WA. The
global conformal group, generated by the modes $L_{-1},L_0,L_{+1}$ of the
\emt\ $T$, puts strong restrictions on these OPEs.

Recall the definition of a quasiprimary operator $\Phi_i$ of dimension $h_i$
(def.\ \ref{def:COPA})\footnote{For convenience, we will drop the hats on
the $\hat{L}$ in the rest of this chapter. }\col
\be \begin{array}{ll}
\L_1 \Phi_i &= 0\\
\L_0 \Phi_i   &= h_i \Phi_i\\
\L_{-1} \Phi_i&= \del \Phi_i\,,\label{eq:QPdef}
\end{array} \ee
where the modes are defined, \vgl{eq:modedef}, as\col
\be
L_n\Phi\equiv [T\Phi]_{n+2}\,.\label{eq:LnonPhi}
\ee
From the definitions in section \ref{sct:WArepresentations}, we see that a
quasiprimary operator furnishes a highest weight representation for the global
conformal algebra.

We will use the following assumption\col
\begin{assumption}\label{ass:QP}
All elements of the OPA are linear combinations of
quasi\-pri\-ma\-ries $\Phi_i$ and their global conformal descendants,
namely their derivatives.
\end{assumption}
This is a natural -- and commonly used -- assumption, but see intermezzo
\ref{int:QPcounterexample}. \puzzle{We defined a \WA\ as a conformal
OPA generated by taking linear combinations, derivatives {\sl and}
composites of quasiprimaries $\{W^i\}$.}

\begin{intermezzo}\label{int:QPcounterexample}%
Consider a free field theory with background charge, see subsection
\ref{ssct:FreeScalar}. Using the definition of the \emt,
\vgl{eq:Tbackgroundcharge}, $T =1/ 2 [\dz X \dz X]_0 - q\,\dz^2 X$, we see
that $\dz X$ is not quasiprimary when $q$ is not zero\col
\[ T \ \dz X\ =\ \opethree{2q}{\dz X}{\dz^2 X} \,.  \]
$X$ is not even a scaling operator\col
\[ T \ X\ =\ \opetwo{q}{\dz X} \,. \]
Hence, including $\dz X$ as a generator of the OPA,
would make the assumption \ref{ass:QP} invalid.

Consider now a vertex operator $V_a$, which is primary with conformal
dimension $h_a=a(a+2q)/2$ with respect to $T$.  Using \vgl{eq:vertexOPE}, we
find for the OPE of $V_a$ with $V_{-a}$\col
\[
V_a(z)\, V_{-a}(w)
= (z-w)^{-a^2} \left(1 +  (z\!-\!w) a\dz X(w) + \ldots\right)
\]
This OPE obeys all associativity conditions.  In particular, this means
that it satisfies Jacobi identities with $T$, given in \vgl{eq:L1onQPOPE},
for all $q$. Clearly, it provides an example of an OPE of two primary
operators which cannot be written in a basis of quasiprimaries (if $q\neq
0$), even not in a basis of highest weight operators.

Of course, by taking a different $T$ as the Virasoro operator, the results of
this chapter can be applied.
\end{intermezzo}

\subsection{Consequences for OPEs}
We will now investigate how this assumption restricts the form of the
OPEs. Consider the OPE of two quasiprimary operators. The
assumption \ref{ass:QP} implies that it can be written as\col
\bea
\Phi_i(z)\Phi_j(w)
&=&\sum_k\sum_{p\geq 0} a_{ij}^k(p)\ \del^p\Phi_k(w) (z-w)^{p-h_{ijk}}\,,
\label{eq:QPOPEpre}
\eea
where we introduced the notation\col
\be
h_{ijk} = h_i+h_j-h_k\,.\label{eq:hijkdef}
\ee
The coefficients $a_{ij}^k(p)$ are (partially) determined by the Jacobi
identities, \vgl{eq:OPEJacRAB}, in the OPA. We now set out to find these
coefficients. The method we use consists in acting with $L_1$ on
\vgl{eq:QPOPEpre}. For $L_1$ acting on the \lhs, we use the Jacobi identities
of $\{T,\Phi_i,\Phi_j\}$, while for the \rhs\ of \vgl{eq:QPOPEpre} we use the
identities of $\{T,T,\Phi_k\}$.

We write the \lhs\ of \vgl{eq:QPOPEpre} with the notation introduced in
\vgl{eq:OPEdef}\col
\bea
\Phi_i(z)\Phi_j(w)&=&\sum_n [\Phi_i\Phi_j]_n (z-w)^{-n}\,.
  \label{eq:QPOPEpre1}
\eea
From the Jacobi identities \re{eq:OPEJacRAB}, or alternatively,
using the commutation rules of the modes of $\Phi_i$ with $L_m$
given in \vgl{eq:PrimaryFieldMode}\footnote{The commutation
relations \vgl{eq:PrimaryFieldMode} were derived for a primary field
$\Phi$. Clearly, they are also valid for a quasiprimary if
$L_m\in\{L_{1},L_0,L_{-1}\}$.}, gives\col \be
L_1{}^m [\Phi_i\Phi_j]_n = (2h_i-m-n)_m
[\Phi_i\Phi_j]_{m+n}\,, \label{eq:L1onQPOPE}
\ee
where the Pochhammer symbol $(a)_n$ is defined in \vgl{eq:Pochdef}.

For the action of $L_1{}^m$ on the \rhs\ of \vgl{eq:QPOPEpre}, we use the
following formula\col
\be
L_1{}^m L_{-1}{}^p\Phi_k = (p-m+1)_m (2h_k+p-m)_{m} L_{-1}{}^{p-m}\Phi_k\,,
\label{eq:L1mL-1nPhi}
\ee
which can be derived using \re{eq:LmL-1n}. Note that we dropped the
term $L_1\Phi_k$ as it is zero.

Comparing the powers of $(z-w)^{-n}$, we find\col
\bea
\lefteqn{(2h_i-m-n)_m \sum_k a_{ij}^k(h_{ijk}-m-n)\
\del^{h_{ijk}-m-n}\Phi_k\ =} \hspace*{4em}\nonumber\\[2mm]
&&\sum_k a_{ij}^k(h_{ijk}-n)\ (h_{ijk}-n-m+1)_m\nonu
&&\qquad  (h_i+h_j+h_k-n-m)_m\ \del^{h_{ijk}-m-n}\Phi_k\,.
\label{eq:aijkeqs}
\eea
We now temporarily assume that the $\{\Phi_i\}$ form an independent set, \ie
no linear combination of the $\Phi_i$ and their derivatives can be made
which is a null operator. This means that the coefficients of all
$\del^p\Phi_k$ in the \lhs\ and \rhs\ have to be equal. Choosing
$m=h_{ijk}\!-\!n$, gives\col
\be
 a_{ij}^k(m)\  m!\ (2h_k)_m\ =\ (h_i-h_j+h_k)_m a_{ij}^k(0)\,.
 \label{eq:QPrecursion}
\ee
Introducing the notation\col
\bea
a_{ij}^k(0)={\cal C}_{ij}^k&&
a_{ij}^k(n)={\cal C}_{ij}^k \QPc{h_i}{h_j}{h_k}{n}\,,
\eea
we conclude that if all $h_k>0$\col
\be
\Phi_i(z)\Phi_j(w) =
\sum_k\sum_{n\geq 0} {\cal C}_{ij}^k \QPc{h_i}{h_j}{h_k}{n}\ \del^n\Phi_k(w)
(z-w)^{n-h_{ijk}}\,, \label{eq:QPOPE}
\ee
where\col
\be
\QPc{h_i}{h_j}{h_k}{n}\ = {(h_i-h_j+h_k)_n \over n!\ (2h_k)_n}\,.
\label{eq:QPcoef}
\ee
Eqs.\ (\ref{eq:QPOPE},\ref{eq:QPcoef}) were derived in \cite{BPZ}. They
enable us to reconstruct the complete OPE when the structure constants ${\cal
C}_{ij}^k$ are given, \ie the coefficients of the quasiprimaries at every
pole.  It is easily verified that the OPEs of the $\Ww_3$-algebra have the
structure given in eqs.\ (\ref{eq:QPOPE},\ref{eq:QPcoef}).

We now treat some special cases where eqs.\ (\ref{eq:QPOPE},\ref{eq:QPcoef})
are not valid.\\
When $h_k$ is half-integer and not strictly positive, \vgl{eq:QPrecursion}
indicates that $a_{ij}^k(1\!-\!2h_k)$ is not determined by the global conformal
transformations. This is because in
this case $\del^{1-2h_k}\Phi_k$ is a quasiprimary (with dimension
$1\!-\!h_k$), as is easily checked using \vgl{eq:L1mL-1nPhi}. This situation
will be mirrored in the next section when considering the full conformal
group.  It occurs because the highest weight representation generated by
$\Phi_k$ for $h_k\leq 0$ is reducible.\\
When null operators occur in the OPA, extra free
coefficients in \vgl{eq:aijkeqs} can appear. However, we can continue to use
eqs.\ (\ref{eq:QPOPE},\ref{eq:QPcoef}) as indeed coefficients if null operators
are arbitrary.

%  First, we notice that, due to
%assumption \ref{ass:QP}, all null operators of non-zero dimension can be
%generated by taking derivatives of null operators with the form
%$\Phi_{n_0}+{\sl derivatives}$, where $\Phi_{n_0}$ is quasiprimary.  Indeed,
%because a correlation function $<X\ \dz\Phi(z)\ Y>$ is equal to $\dz_z<X\
%\Phi(z)\ Y>$, $\dz\Phi$ cannot be a null operator unless either $\Phi$ is
%null or $\Phi$ has dimension zero.  \puzzle{Dit schijnt te zeggen dat
%$\del^{1-2h_k}\Phi_k$  geen null operator kan zijn ??}\\ This means, if we
%simply discard $\Phi_{n_0}$ from the set of quasiprimaries $\{\Phi_i\}$ we
%used in \vgl{eq:QPOPEpre}, assumption \ref{ass:QPindependent} is restored.

\subsection{Finding the quasiprimaries in an OPE\label{ssct:QPinOPE}}
For practical applications, the quasiprimaries at each pole have to be
identified when the complete OPE is given. This is the reverse problem of
the previous subsection.

We write the quasiprimary at the $m$-th
order as $QP^m(\Phi_i,\Phi_j)$. It can be determined in a recursive way by
using the results of the previous subsection. Indeed, the highest pole of
the OPE is by assumption \ref{ass:QP} quasiprimary. We can substract the
complete ``tower'' of derivatives of this quasiprimary from the OPE to end up
with a new Laurent series where the most singular term is again quasiprimary.
We find\col
\bea
\leqn{QP^m(\Phi_i,\Phi_j)\ =\ [AB]_m}
&&\!\!- \sum_{n\geq 1}\!
\QPc{h_i}{h_j}{h_i\!+\!h_j\!-\!m\!-\!n}{n} \ \del^n
(QP^{m+n}(\Phi_i,\Phi_j))\,.\label{eq:QPqdefrecursive}
\eea
In particular, for $m=0$ this formula provides a definition of a
quasiprimary normal ordered product of two quasiprimaries, for example\col
\be
{\cal NO}(T\Phi_i) = [T\Phi_i]_0 - {3\over 2 (2h_i+1)} \del^2 \Phi_i\,,
\ee
of which \vgl{eq:Lambdadef} is a special case. More general, we see that
$QP^m(\Phi_i,\Phi_j)$ for $m\leq 0$ are composite
quasiprimaries. The formula \re{eq:QPqdefrecursive} appears also in
\cite{blumetal}.

The recursive definition \vgl{eq:QPqdefrecursive} of the operators
$QP^m$ is a very inefficient, and complicated, way of computing the
quasiprimaries. To simplify this definition, we note that it can be rewritten
as\col
\be QP^m(\Phi_i,\Phi_j) = \sum_{n\geq 0} \QPs^m_n(h_i,h_j)\
    \del^n [\Phi_i\Phi_j]_{n+m}\,, \label{eq:QPqdef}
\ee
where the coefficients $\QPs^m_n(h_i,h_j)$ are determined by
\vgl{eq:QPqdefrecursive}. We find\col
\be
\sum_{k=0}^n
\QPs^{m+k}_{n-k}(h_i,h_j)\QPc{h_i}{h_j}{h_i\!+\!h_j\!-\!m\!-\!k}{k} = 0\,.
\label{eq:findQPa}
\ee
We will provide a closed form for the solution of \vgl{eq:findQPa} by
requiring that \vgl{eq:QPqdef} defines a quasiprimary operator\col
\be
L_1\ QP^m(\Phi_i,\Phi_j)=0\,.\label{eq:L1onQPm}
\ee
 To do this, we need the action of $L_1$ on all the
terms in \vgl{eq:QPqdef}\col
\bea
\leqn{ L_1(\del^n[\Phi_i\Phi_j]_p)\ =\
 (2h_i-p-1)\del^n[\Phi_i\Phi_j]_{p+1}\ +}
&& n(2h_i+2h_j-2p+n-1)\del^{n-1}[\Phi_i\Phi_j]_p\,,
\label{eq:L1L-1monpole}
\eea
which can be derived by commuting $L_1$ to the right. The result
contains a term similar to \vgl{eq:L1mL-1nPhi}, and a term coming
from the action of the $L_1$ on $[\Phi_i\Phi_j]_p$, see
\vgl{eq:L1onQPOPE}. We find for \vgl{eq:L1onQPm}\col
\bea
\leqn{\sum_{n\geq 1}\del^{n-1}[\Phi_i\Phi_j]_{m+n}
\biggl( \QPs^m_{n-1}(h_i,h_j)\,(2h_i-n-m)\ +}
&&\qquad  \QPs^m_n(h_i,h_j)\, n\,(2h_i+2h_j-2m-n-1)\biggr)\ =\ 0\,.
\label{eq:QPfromOPEequation}
\eea
In the general case, this amounts to a recursive relation\footnote{Of course,
when $\del^{n-1}[\Phi_i\Phi_j]_{m+n}$ is proportional (up to null operators) to
$\del^n[\Phi_i\Phi_j]_{m+n+1}$, \vgl{eq:QPfromOPEequation} fixes only a
combination of $\QPs^m_n$ and $\QPs^m_{n+1}$, but it is this combination that will
appear in \vgl{eq:QPqdef}.}\col
\be
\QPs^m_n(h_i,h_j)\ =\
-{2h_i-n-m \over  n(2h_i+2h_j-2m-n-1)} \QPs^m_{n-1}(h_i,h_j)\,.
\label{eq:QPqcoefrecursive}
\ee
Choosing $\QPs^m_0$ equal to $1$, gives\col
\be
\QPs^m_n(h_i,h_j)\ =\ (-1)^n {(2h_i-n-m)_n \over  n!(2h_i+2h_j-2m-n-1)_n}\,.
\label{eq:QPqcoef}
\ee
\begin{intermezzo}
We can directly check that the coefficients given in \vgl{eq:QPqcoef} indeed
satisfy \vgl{eq:findQPa}. This involves the summation of terms
which are a product of factorials. This summation, and most of the other sums
that will be used in this chapter, can be done using the {\tt
Algebra`SymbolicSum`} package of \Mathematica, which provides an
implementation of the Gosper algorithm. However, the current version of
this package does not handle the {\tt Pochhammer} function. This can be
remedied by converting the Pochhammer functions to a quotient of
$\Gamma$-functions, \vgl{eq:Pochdef}. The results returned by {\tt
SymbolicSum} in the case at hand contain a cosecans. This can again be
converted to $\Gamma$-functions using the identity\col
\[  \sin(\pi x) = {\pi\over \Gamma(x) \Gamma(1-x)}\,.
\]
The result then contains terms like $\Gamma(-n)$ which are infinite for
positive $n$. However, as all sums in this chapter are finite, these
infinities necessarily disappear against terms like $\Gamma(2-n)^{-1}$.
It is possible to write a set of \Mathematica\ rules which checks this
cancellation automatically by converting quotients of $\Gamma$-functions
back to Pochhammer symbols. All sums in this chapter are computed in this
way.

In the case at hand, \vgl{eq:findQPa}, the final result contains a
factor\col
\[ -2\,n\,{\sqrt{\pi }}\,{\Gamma}(2\,n) +
  {4^n}\,{\Gamma}({1\over 2} + n)\,{\Gamma}(1 + n)\,,\]
which is zero (except at the singularities), proving that \vgl{eq:QPqcoef}
provides the solution of \vgl{eq:findQPa}.
\end{intermezzo}

As the operator $QP^m(\Phi_j,\Phi_i)$ extracts the quasiprimary at the $m$-th
order pole, we expect that reversing $i$ and $j$ does not give a new operator.
Indeed, an explicit calculation gives\footnote{Ref.~\cite{blumetal} contains
this formula  for $m=0,-1$.}\col
\be
QP^m(\Phi_j,\Phi_i) = (-1)^{ij + m}\ QP^m(\Phi_i,\Phi_j)\,,
\ee
where $ij$ in the phase factor gives a sign depending on the parity of
the quasiprimaries ($-1$ if both are fermionic, $1$ otherwise). This
equation can be proven using \vgl{eq:OPEcomm} and the identity\col
\be
\sum_{m=0}^n {(-1)^m \over (n-m)!} \QPs^q_m(h_j,h_i) = \QPs^q_n(h_i,h_j)\,.
\ee
%Also, due to the Jacobi identities for OPEs, \vgl{eq:OPEJacRAB}, a relation
%for the reordering of\col
%\[QP^m(\Phi_i,QP^n(\Phi_j,\Phi_k))\]
%exists. It would be useful for calculating OPEs directly in a basis of
%quasiprimaries.

To conclude this subsection, let us discuss a special case where
\vgl{eq:QPqcoef} is not valid, namely when $n\geq n_c$, where we define\col
\be
n_c=2(h_i\!+\!h_j\!-\!m)-1\,.
\ee
We observe that the coefficient $\QPs^m_{n_c}(h_i,h_j)$ is not
determined by \vgl{eq:QPfromOPEequation}. On the other hand, \vgl{eq:QPqdef}
shows that for such $n\geq n_c$, the $\QPs^m_n$ are coefficients of the $n$-th
derivative of a operator with dimension $h\leq 0$. As discussed in
the previous subsection,for any quasiprimary of dimension $h_k\leq 0$,
$\del^{1-2h_k}\Phi_k$ is also quasiprimary. This is clearly the origin of the
freedom in  $\QPs^m_{n_c}(h_i,h_j)$.\\
Let us take as an example $n=n_c$, and assume that there are no poles of
order higher than $m+n_c$. In this case $[\Phi_i\Phi_j]_{m+n_c}$ is
a quasiprimary of negative or zero conformal dimension. In this case,
$QP^m(\Phi_i, \Phi_j)$ is indeed quasiprimary for arbitrary
$\QPs^m_{n_c}(h_i,h_j)$.

\section{Consequences of the full conformal group
\label{sct:localConformalRestrictions}}
In the previous section, the global conformal transformations and their
implications on OPEs were studied. Here, we extend the analysis to
include all modes $L_n$ of the \emt\ $T$, acting on primary operators. A
primary operator $\Psi_i$ of dimension $h_i$ satisfies (def.
\ref{def:COPA})\col
\be \begin{array}{lll}
L_{n} \Psi_i&= 0&\qquad n>0\\
L_0 \Psi_i &= h_i&\\
L_{-1} \Psi_i &= \del \Psi_i\,.&
\end{array}\label{eq:Pdef}
\ee
Similarly to the previous section, we use the following assumption.
\puzzle{zie ook Stany, HFW -> primary}
\begin{assumption}\label{ass:P}
All elements of the OPA are linear combinations of the pri\-ma\-ries
$\Psi_i$ and their Virasoro descendants.
\end{assumption}
Note that we included the unit operator $\unity$ in the list of primaries
which generate the OPA. The \emt\ $T$ is then a Virasoro descendant of
$\unity$\col
\be
L_{-2}\unity = T\,.
\ee

\subsection{Restrictions of conformal covariance on the OPEs}
As in the previous section, assumption \ref{ass:P} implies that the OPE of two
primary operators can be written as\col
\be
\Psi_i(z)\Psi_j(w) =
\sum_k\sum_{\{p\}} C_{ij}{}^k\ \Pc{h_i}{h_j}{h_k}{\{p\}}\
L_{-\{p\}}\Psi_k(w)\ (z-w)^{P-h_{ijk}}\,, \label{eq:POPE}
\ee
where the sum is over all ordered sequences $\{p\}$ and $P$ is
the level of $\{p\}$ (see section
\ref{sct:WArepresentations} for notations). To rewrite \vgl{eq:POPE} in
terms of composites with $T$, we use \vgl{eq:Poleregdef}\col
\bea
L_{-p} \Psi = [T\Psi]_{2-p} = {1\over (p-2)!} [\del^{p-2}T\ \Psi]_0
&&p\geq 2\,.
\eea
The $\beta$-coefficients defined in \vgl{eq:POPE} depend only on the
dimensions of the primaries. They were introduced in \cite{BPZ}, where the
first two levels where computed explicitly. The equations which
determine these coefficients where explicitly written down in
\cite{josestany:W26} using the inner product given in section
\ref{sct:WArepresentations}.  We now show how these equations can be derived
independently of the inproduct, in a way entirely analogous to the previous
section.

We will act with positive modes $L_n$ on \vgl{eq:POPE}. For the \lhs,
the Jacobi identities \re{eq:OPEJacRAB} with $T$ imply\col
\be
L_p [\Psi_i\Psi_j]_n\ =\ h_j (p-1) [\Psi_i\Psi_j]_n +
   [(L_{-1}\Psi_i)\Psi_j]_{n+1}\,.\label{eq:LnonHWOPE}
\ee
Because $\Psi_i$ is a primary operator, we have
that $L_{-1}\Psi_i=\del\Psi_i$,  \vgl{eq:Pdef}, which gives together with
\vgl{eq:PolederL}\col \be
 L_p [\Psi_i\Psi_j]_{h_{ijk}-P} = (h_i p-h_j+h_k+P-p)
[\Psi_i\Psi_j]_{h_{ijk}-P+p}\,. \label{eq:LnonPOPE}
\ee
Extending this result to a partition $\{p\}$ of $P$, we find\col
\be
L_{\{p\}}\left( [\Psi_i\Psi_j]_{h_{ijk}-P}\right) = f(h_i,h_j,h_k,\{p\})
[\Psi_i\Psi_j]_{h_{ijk}}\,, \label{eq:LpartitiononPOPE}
\ee
where
\be
f(h_i,h_j,h_k,\{p\})\equiv
\prod_l\left(h_i p_l-h_j+h_k+\sum_{l'>l}p_{l'}\right)\,.
\label{eq:Pfdef}
\ee

On the other hand, the action of $L_{\{p\}}$ on the \rhs\ of \vgl{eq:POPE} can
be computed by commuting the positive modes to the right, where they
annihilate $\Psi_k$.  The result depends on the dimensions of the fields,
but also on the central charge $c$ appearing in the commutators
of the Virasoro algebra, \vgl{eq:VirasoroMode}.  No easy formula can be
given for the resulting expression, we write\col
\be
L_{\{m\}}L_{-\{n\}}\Psi_k = S^{\{m\},\{n\}}(h_k,c)\ \Psi_k\,,
\ee
for partitions of the same level N.
The matrix $S$ is the \v{S}apovalov form defined in section
\ref{sct:WArepresentations}. The dimension of $S$ is given by the number
of (ordered) partitions of $N$, which we denote by $p(N)$.
\begin{intermezzo}
As an example we give the matrix $S$ at level $2$\col
\[
\begin{array}{rl|rl}
L_2L_{-2}\Psi&=\ (4h+c/2)\Psi&  L_2L_{-1}{}^2\Psi&=\ 6h\Psi\\
L_{1}{}^2L_{-2}\Psi&=\ 6h\Psi&  L_{1}{}^2L_{-1}{}^2\Psi&=\ 4h(2h+1)\Psi\,,
\end{array}
\]
where $\Psi$ is a primary with dimension $h$. $S$ is a symmetric matrix.
This follows from the fact that the Virasoro commutators
\re{eq:VirasoroMode} are invariant under the substitution $L_n\rightarrow
L_{-n}$.
\end{intermezzo}
Assuming independence of the Virasoro descendants, we get\col
\be
\sum_{\{m\}} S^{\{n\},\{m\}} \Pc{h_i}{h_j}{h_k}{\{n\}}  \ =\
f(h_i,h_j,h_k,\{n\})\,,\label{eq:Pcequation}
\ee
where the sum is over all partitions of $N$. This equation determines the
$\beta$-coefficients at level $N$ completely, unless the matrix $S$ is
singular. The singular vectors of $S$ correspond to primary Virasoro
descendants\puzzle{, see lemma \ref{lem:HWOdesc}}. Clearly, their
coefficients cannot be determined by the Jacobi identities with $T$. This
corresponds to the poles in the $\beta$-coefficients.
\begin{intermezzo}\label{int:betalevel2}%
From the previous intermezzo, we can give the $\beta$-coef\-fi\-cients at
level two\col
\beastar
\Pc{h_i}{h_j}{h_k}{\{1,1\}}&=&
 \Bigl(c\left( h_i - h_j + h_k \right)
      \left( 1 + h_i - h_j + h_k \right)  +\\
&&\quad     4h_k\bigl( -4h_i + 2{{h_i}^2} + h_j - 4h_ih_j +
        2{{h_j}^2} -\\
&&\quad\quad   h_k + 4h_ih_k - 4h_jh_k +  2{{h_k}^2} \bigr) \Bigr) \Big/\\
&&\bigl(4h_k\left( c\left( 1 + 2h_k \right)  +
   2h_k\left( -5 + 8h_k \right)  \right) \bigr)\\
\Pc{h_i}{h_j}{h_k}{\{2\}}&=&
\bigl(h_i - 3{{h_i}^2} + h_j + 6h_ih_j - 3{{h_j}^2} - h_k\\
&&\quad  + 2h_ih_k + 2h_jh_k + {{h_k}^2}\bigr) \Big/\\
&&   \bigl(c\left( 1 + 2h_k \right)  +
     2h_k\left( -5 + 8h_k \right) \bigr).
\eeastar
These expressions are already complicated.  The coefficient
of $L_{-1}{}^2\Psi_k$ is undetermined when $h_k=0$ because
$\dz\Psi_k$ is a primary of dimension $1$ in this case. Also, when
\[
c={{2\,\left( 5 - 8\,h_k \right) \,h_k}\over {1 + 2\,h_k}}\,
\]
the quasiprimary combination\col
\[
[T \Psi_k]_0 - {3\dz^2\Psi_k\over 2(1 + 2h_k)}
\]
turns out to be primary, which explains the second pole in the
$\beta$-coef\-fi\-cients.
\end{intermezzo}

When trying to work out the $\beta$-coefficients at higher level, the main
problem lies in solving \vgl{eq:Pcequation}. Standard numerical
algorithms cannot be used, as the matrix $S$ contains the constant $c$, which
we want to keep as a free parameter. Also, the dimension $p(N)$ of the
matrix increases rapidly with the required level, which makes solving
\vgl{eq:Pcequation} for high levels unfeasible. In the next subsection,
we will work in a basis of quasiprimaries. This will increase the
complexity of computing $S$, but brings it to block-diagonal form, making
the solution of equation \vgl{eq:Pcequation} much easier. Also, the
primary descendants are of course necessarily quasiprimaries, so they are
much easier to find in this basis.

We wish to stress that this analysis for Virasoro-primary operators cannot be
extended to highest weight operators.  Indeed, to determine the coefficients
of the Virasoro descendants in an OPE, we explicitly used extra information
about $L_{-1}$ on a primary operator, namely that $L_{-1} \Psi = \partial
\Psi$ (see \vgl{eq:LnonHWOPE}).  As we do not have such information at
present for nonlinear algebras, it is not yet possible to write down
$\Ww$-covariant OPEs.  See also section 4.1.1 in \cite{Stany:thesis}.

\subsection{Virasoro descendants in a quasiprimary basis\label{ssct:VirQP}}
Our aim in this subsection is to use the information of section
\ref{sct:globalConformalRestrictions} to bring the system of equations in
\vgl{eq:Pcequation} in a block-diagonal form. This will be done by changing
to a basis of quasiprimary Virasoro descendants and their derivatives. This
basis is defined in terms of operators $\tilde{L}_m$ and $L_{\pm1}$,
where\col
\be
\tilde{L}_m \Psi\equiv QP^{m+2}(T,\Psi)\,,\label{eq:Pmdef}
\ee
for $\Psi$ quasiprimary (see eqs.\
\re{eq:QPqdef} and \re{eq:QPqcoef})\footnote{These operators were also
defined in \cite{kauwat}, but apparently not used.} The definition of
$\tilde{L}_m$ depends on the dimension of the operator on which it is acting.
This can be avoided by using $L_0$. However, in the normalisation we chose,
one needs to introduce factors $(2L_0+i)^{-1}$. To avoid this, we will write
$\tilde{L}_m(h)$ when confusion can arise.

The action of $\tilde{L}_m$ is well-defined for any quasiprimary $\Phi$. When
$m<-1$, the sum in \vgl{eq:QPqdef} contains only a finite number of terms
because $\QPs^{m+2}_n(2,h_j)=0$ for $n\geq 2\!-\!m$.
When $m>1$, $L_{n+m}\Phi=[T\Phi]_{n+m+2}$ is zero for $n$ large enough. In
particular, when $\Phi$ is a level $N$ descendant of a primary operator,
$L_{n+m}\Phi=0$ if $n+m>N$.

We will denote partitions which do not contain the number $1$ as
$\{\tilde{n}\}$, and write $p_1(N)$ for the number of such partitions.
Note that all partitions of level $N$ which do contain a $1$ can be
obtained by adding a $1$ to the normal partitions of level $N-1$. This
implies $p_1(N)=p(N)-p(N-1)$.

The descendants
\be
L_{-1}{}^{N-N_k}\tilde{L}_{-\{\tilde{n}_k\}}\Psi_j
\ee
(where the partition $\{\tilde{n}_k\}$ has level $N_k$) for $N_k=0,2,3,\ldots
N$ span the space of all Virasoro descendants at level $N$. When no primary
descendants exist, they are independent.

Once we know the coefficients of the quasiprimaries at all levels $M\leq
N$, we can use the results of the previous section to find the complete
OPE\col
\bea
\leqn{\Psi_i(z)\Psi_j(w) =
\sum_k\sum_{\{\tilde{n}\}} C_{ij}{}^k\ \Pqpc{h_i}{h_j}{h_k}{\{\tilde{n}\}} }
&&\sum_{m\geq 0}\QPc{h_i}{h_j}{h_k+N}{m}\ \del^m
\tilde{L}_{-\{n\}}\Psi_k(w)\ (z-w)^{N+m-h_{ijk}}\,. \label{eq:PqpOPE}
\eea
To determine the $\tilde{\beta}$-coefficients, we proceed as
before, after a projection on the quasiprimaries in \vgl{eq:PqpOPE}.
We act with a sequence of $\tilde{L}_m$ operators, $m\geq 2$, on
\vgl{eq:PqpOPE}. For the \rhs, we need the matrix $\tilde{S}$ which
appears in\col
\be
\tilde{L}_{\{\tilde{m}\}}\tilde{L}_{-\{\tilde{n}\}}\Psi_k =
    \tilde{S}^{\{\tilde{m}\},\{\tilde{n}\}}(h_k,c)\ \Psi_k\,,
\ee
To compute $\tilde{S}$, we have to know the commutation rules for the
$\tilde{L}_n$ operators. We will need only the case where we commute a
positive mode through a negative mode. We find for $m, n>1$\col \bea
\leqn{\tilde{L}_{m}(h+n)\tilde{L}_{-n}(h)\ =\
   \delta_{m-n} f_1(h,m) + \tilde{L}_{m-n}f_2(h,m,-n) }
&&+\sum_{p\geq 2-\min(n,m)}
       \tilde{L}_{-n-p}(h\!-\!m\!-\!p)\tilde{L}_{m+p}(h)\ f_3(h,m,-n,p)\,,
\label{eq:Pcomrules}
\eea
where we take $\tilde{L}_{\pm 1}=0$ and $\tilde{L}_0=L_0$. The
coefficients $f_i$ appearing in \vgl{eq:Pcomrules} are quite involved and are
given in appendix \ref{sct:appCommutators}, see eqs.\ (\ref{eq:Ltf1f2},
\ref{eq:Ltf3}). The sum in
\vgl{eq:Pcomrules} contains an infinite number of terms, but we have to keep
only the terms with $p+m\leq N$ when acting on a Virasoro descendant of
level $N$.

Applying $\tilde{L}_{\{\tilde{m}\}}$ to the \lhs\ of \vgl{eq:PqpOPE}, after
projecting on the quasiprimaries, we get the analogue of
\vgl{eq:LpartitiononPOPE}\col \be
\tilde{L}_{\{\tilde{n}\}}\left( QP^{h_{ijk}-\tilde{N}}(\Psi_i,\Psi_j)\right) =
 \tilde{f}(h_i,h_j,h_k,\{\tilde{n}\})
QP^{h_{ijk}}(\Psi_i,\Psi_j)\,, \label{eq:PpartitiononPOPE}
\ee
where the $\tilde{f}(h_i,h_j,h_k,\{\tilde{n}\}$ are given in the appendix,
 \vgl{eq:Pftdef}.

We have the final equation\col
\be
\sum_{\{\tilde{m}\}} \tilde{S}^{\{\tilde{n}\},\{\tilde{m}\}}
   \Pqpc{h_i}{h_j}{h_k}{\{\tilde{n}\}}  \ =\
\tilde{f}(h_i,h_j,h_k,\{\tilde{n}\})\,,\label{eq:Pqpcequation}
\ee
where the sum is over all partitions of $N$, not containing the number $1$.

\begin{intermezzo}
As an example, at level $2$ we find\col
\be
\tilde{S}^{\{2\},\{2\}}(h,c)\ =\
   {{c - 10h + 2 c h + 16 h^2}\over {2 \left( 1 + 2 h \right) }}\,.
\ee
This gives for $\Pqpc{h_i}{h_j}{h_k}{\{2\}}$ immediately the result of
intermezzo \ref{int:betalevel2}. In contrast to when using the
$L_{-\{n\}}$ basis, no problem of inverting $\tilde{S}$ occurs when we
consider a dimension $0$ operator\footnote{
$\tilde{f}(h_i, h_j, h_j-h_k-2,\{2\})$ as given in \vgl{eq:Pftdef} does 
have an
apparent factor $h_k$ in the denominator, but this is a consequence of
writing $\tilde{f}$ in a closed form. No factor $h_k^{-1}$ appears in the
definition of $\tilde{f}(h_i, h_j, h_j-h_k-2,2)$, see \vgl{eq:f5def}.
Indeed, the factor
$h_k^{-1}$ disappears after factorisation.}. We find\col
\be
\Pqpc{h_i}{h_j}{0}{\{2\}} =
{h_i - 3h_i^2 + h_j + 6h_i h_j - 3h_j^2 \over c}\,,
\ee
which reduces to $2h_i/c$ for $h_j=h_i$.

A particular example of a
dimension zero operator is the unit operator $\unity$. We see that the
conventional normalisation of a primary operator of dimension $h$ with OPE
$\Psi(z)\,\Psi(w)=c/h (z-w)^{-2h}+ \ldots$ gives as level two descendant
simply $2T$, \ie with a $c$-independent coefficient.
\end{intermezzo}

The complexity of the coefficients in eqs.\ \re{eq:Pcomrules} and
\re{eq:PpartitiononPOPE} makes these formulas unsuited for pen and paper
calculations, but is no problem when using a symbolic
manipulation program. As discussed in intermezzo \ref{int:LtildeTimings},
it turns out that the calculation of $S$ takes even more time than needed
for $\tilde{S}$. Actually, we are more interested in the quasiprimary
basis, so we do not need $S$ at all. Even more important for computer
implementation \cite{OPEconf} is that we reduced the dimension of the
system of equations \vgl{eq:Pqpcequation} from $p(N)$ to
$p_1(N)=p(N)-p(N-1)$.\puzzle{Compare growth asymptotically somehow}
That this is a significant simplification is easily seen on a few examples.
For levels $2$ through $8$, $p(N)$ is $\{2, 3, 5, 7, 11, 15, 22\}$ while
$p_1(N)$ is $\{1, 1, 2, 2, 4, 4, 7\}$.
\begin{intermezzo}\label{int:LtildeTimings}%
As an illustration of the above arguments, we will present some timings
when using an implementation in \Mathematica\ of the formulas in this and
the previous section. Timings were done in \Mathematica\ 2.2 running on a
$486$ ($50$ MHz).\\
To compare the computation of the $S$ and $\tilde{S}$ matrices, we compute
the case of an OPE where descendants of a primary occur
up to level $7$, \ie we need the matrices for level $1$ to $7$. As the
matrices are computed recursively, we use an algorithm that stores and
reuses its results. For $S$, the total time needed is $20s$, while for
$\tilde{S}$ it is only $4s$. For level $9$, the timings become
$110s$ and $40s$ respectively. When going to even higher level, the
difference in timing decreases. However, for the most complicated algebras
explicitly constructed up to now (\eg $\Ww\!A_5$ in \cite{Klaus:WA5}) the
maximum level of a descendant of a primary (not equal to the unit operator)
is $8$.

The advantage of using the quasiprimary basis shows up even more when
solving the equations for the $\beta,\tilde{\beta}$-coefficients, eqs.\
\re{eq:Pcequation} and \re{eq:Pqpcequation}. We computed the coefficients
for $h_i=6,h_j=7,h_k=10$ and $c$ arbitrary. Although the equations we need
to solve are linear equations, serious problems occured for moderately
high level.\\
The \Mathematica\ (version 2.2 or lower) function
{\tt Solve} has a serious deficiency in that it does not simplify
intermediate results. This means that when arbitrary constants are present
in the equations, the solution is
usually a quite big expression which simplifies drastically after
factorisation. Unfortunately, this factorisation can take a very long
time. For even more complicated equations {\tt Solve} runs into problems
because the intermediate results grow too fast. K. Hornfeck and I wrote a
separate package to solve linear equations using Gaussian elimination,
simplifying the coefficients of the variables at each step.\\
We find the following CPU-times for the solution of the equations
\re{eq:Pcequation} and \re{eq:Pqpcequation}.

\begin{tabular}{|c||r|r||r|r||}\hline
level
& $\beta$ with {\tt Solve}& $\beta$ with {\tt LinSolve}
& $\tilde{\beta}$ with {\tt Solve}&$\tilde{\beta}$ with {\tt LinSolve}
\\ \hline
4 &            6.3s\ \ &   4.7s\ \ &        1.6s\ \ &  1.9s\ \ \\
5 & *         18.4s\ \ &  14.5s\ \ &        1.6s\ \ &  1.9s\ \ \\
6 & *\      $>400$s\ \ &  69.6s\ \ & *\    15.3s\ \ & 54.6s\ \ \\ \hline
\end{tabular}

\noindent
In this table, a star means that factorisation of the result
of {\tt Solve} did not succeed in a reasonable amount of time.
\end{intermezzo}

As before, \vgl{eq:Pqpcequation} only determines the
$\tilde{\beta}$-coefficients when $\tilde{S}$ is nonsingular.  Clearly,
any singular vectors of $\tilde{S}$ correspond to primary descendants,
whose coefficients remain unfixed by Jacobi identities with $T$. These
free coefficient will appear automatically when solving the linear
equations.

\subsection{Virasoro descendants of the unit
operator\label{ssct:descendantsOne}}
We now consider the case of the Virasoro descendants of the unit operator,
which is a primary of dimension zero. In this case, it turns out that the
matrix $\tilde{S}$ has a large amount of singular vectors, even for general
$c$. We usually need the descendants of $\unity$ to a much higher level,
because most \WA s studied up to now have primary generators with dimension
larger than $2$. This means we should reduce the number of (linear) equations
in \vgl{eq:Pqpcequation} as much as possible when computing
$\QPc{h_i}{h_j}{0}{m}$.

The  singular vectors of $\tilde{S}$ for a dimension zero field are in
general primary fields. For the unit operator however, the singular vectors
are exactly zero. There is thus no use in keeping them in the calculation.
This provides an additional reason for studying the singular vectors of
$\tilde{S}$ for dimension zero.

Some examples of these singular vectors are easily constructed. Obviously we
have that $L_{-1}\unity=\del\unity=0$. $\tilde{L}_{-n}\unity$ is zero for
$n>2$ as well. Indeed, the definition \re{eq:LnonPhi} shows that
$\tilde{L}_{-n}\unity$ is proportional to $\del^{n-2}T$, which is clearly not
quasiprimary unless $n=2$, hence it is zero for $n\neq 2$. One can also
prove that $\tilde{L}_{-n}T$ is zero for odd $n$.
More complicated relations at higher level exist.

Before discussing how to find a basis for the quasiprimary descendants of
$\unity$, let us determine the number of independent quasiprimaries at
level $N$, which we will denote as $p_2(N)$. First, the number of
independent descendants is simply $p_1(N)$. Indeed, when ordering the
$L_{-\{n\}}$ as before, the $L_{-1}$ will act on $\unity$ first, giving
zero. Hence, all partitions containing $1$ should be dropped.
For general central charge, no further relations between the remaining
descendants exist. To compute the number of quasiprimary descendants of
$\unity$ at level $N$, we take all descendants, and ``substract" those
which are derivatives of the descendants at the previous level, \ie
$p_2(N) = p_1(N) - p_1(N-1)$.

We can use this information to find other singular vectors than the ones
given already. Because $p_2(7)=0$, there are no descendants at level $7$,
which means that $\tilde{L}_{-3}\tilde{L}_{-2}\tilde{L}_{-2}\unity$ is zero,
which can be checked explicitly. At level $9$, there is only one independent
quasiprimary. Eliminating the zeroes we found at previous levels, two
candidates remain. We find\col
\be
\tilde{L}_{-\{3,2,2,2\}}\unity = -{8\over 5} \tilde{L}_{-\{5,2,2\}}\unity\,.
\label{eq:descOnelevel9}
\ee
Similar relations exist at higher (not necessarily odd) levels.

Remembering the rule for the number $p_2(N)$ of independent quasiprimary
descendants of $\unity$, we propose\footnote{
This conjecture was established together with K.~Hornfeck after suggestions
of M.~Flohr. In \cite{Flohr:thesis}, the same conjecture is made, but
not in detail. Actually, \cite{Flohr:thesis} claims to have proved the
conjecture, but his argument seems to be solely based on counting the number
of independent descendants.\puzzle{}}\col
\begin{conjecture}
The partitions which give the independent descendants of the unit operator
at level $N$ can be found as follows\col
\begin{itemize}
\item Take the partitions (not containing the number $1$) at level $N-1$
and order them in increasing lexicographic order ($\{32\}\,<\,\{33\}$).
\item ``Increment'' these partitions one by one, such that
 no partition of level $N$ is obtained twice.
\item The Virasoro descendants of $\unity$ at level $N$ correspond to
partitions at level $N$ which are not in the
list constructed in the previous step.
\end{itemize}
\end{conjecture}
With ``incrementing'' a partition $\{n_1,n_2,\ldots n_k\}$, we mean adding a
$1$ at a position $i$ where $n_{i-1}\geq n_i+1$. As an
example, incrementing $\{3,2,2\}$ gives $\{3,3,2\}$ or $\{4,2,2\}$.

The first nontrivial example of this procedure is at level $9$. At level $8$
the partitions are $ \{2222\ 332\ 44\ 53\ 62\ 8\}$.  Incrementing the first
three partitions gives $\{3222\ 333\ 54\}$. We now have to increment the
partition $\{53\}$. The partition $\{54\}$ is already in  our list, so we
should take $\{63\}$. In the next steps $\{72\}$ and $\{9\}$ are found. The
only partition of level $9$ that remains is $\{522\}$, in agreement with
\vgl{eq:descOnelevel9}.

This conjecture was checked up to level $16$.

\subsection{Finding the primaries in an OPE\label{ssct:PinOPE}}
Entirely analogously to \vgl{eq:QPqdefrecursive}, we can define\col
\bea
\leqn{P^m(\Psi_i,\Psi_j) = QP^m(A,B) -}
&&\!\!\sum_{\{\tilde{n}\}}
\Pqpc{h_i}{h_j}{h_i\!+\!h_j\!-\!m\!-\!N}{\{n\}} \ \tilde{L}_{-\{\tilde{n}\}}
P^{m+N}(\Phi_i,\Phi_j)\,.\label{eq:Pqdefrecursive}
\eea
This can be rewritten as\col
\be
P^m(\Psi_i,\Psi_j) = \sum_{\{\tilde{n}\}} \QPs^m_{\{\tilde{n}\}}(h_i,h_j)\
    \tilde{L}_{-\{\tilde{n}\}} QP^{m+N}(\Psi_i,\Psi_j)\,,
\label{eq:Pqdef} \ee
where the sum is over all partitions of level not equal to one.
We have not found a closed form for this expression. Even determining the
recursion relations for the coefficients $\QPs^m_{\{\tilde{n}\}}(h_i,h_j)$
seems unfeasible in general.

In many cases, the action of $P^m$ will give zero, as there simply is no
primary at a certain pole.

The operators $P^m$ are particularly convenient to construct primary
composites. Indeed, for $m\geq 2$, the operator $P^{-m}(\Psi_i, \Psi_j)$
contains $[\Psi_i\Psi_j]_{-m}\sim [\del^m \Psi_i\ \Psi_j]_0$. In
\cite{Verstegen:bootstrapinf}, a formula is given to count the number of
primaries that can be constructed\puzzle{}.

\section{An overview of \WA s
\label{sct:WAclassification}}
In this section, we present a overview on the \WA s which are known.
For a more in-depth review, see \cite{WAreviewpbks}, which we also follow for
the nomenclature of the \WA s.

We first distinguish different classes of \WA s. ``Deformable" algebras
are algebras that exist for generic values of the central charge.
\WA s which are only associative for specific values of $c$ are called
non-deformable. In a "freely generated" \WA, for generic $c$ no relations
exist between the generators, \ie no null operators appear.

Up to now, all efforts have been concentrated on algebras in the following
class\col
\begin{assumption}
\begin{itemize}
\item The OPA is generated by a set of primary operators and the Virasoro
operator.
\item All generators have strictly positive conformal dimension.
\item The unit operator occurs only in OPEs
between primaries of the same conformal dimension.
\item Generators which are null operators are discarded.
\end{itemize}\label{ass:normalWAs}%
\end{assumption}
Not all $\Ww$-algebras of interest fall within this class. For instance,
it was recently shown \cite{Sergey:linearWA} that the (nonlinear) $\Ww_3$
algebra and the Bershadsky algebra \cite{PolyBosSusy,mic} with bosonic
generators of dimensions $(2,3/2,3/2,2)$ can be viewed as subalgebras of
{\sl linear} algebras with a null operator as generator, and a
non-primary dimension $1$ operator. Yet even with the restrictions
\ref{ass:normalWAs}, no complete classification has been found yet, see
\cite{WAlist}.

The current status in our knowledge of \WA s can be compared to the study of
Lie algebras before Cartan presented his classification. Different
construction methods of \WA s are known, giving rise to series of \WA s with
common features. In a sense, they are analogous to the ``classical Lie
algebras'', which were also known via a realisation. A few isolated cases
have also been constructed, although they currently seem to fit in the general
pattern.

We will now
comment on some of the more important construction methods and end with the
(incomplete) classification proposed in \cite{Bowcockcontraction}.

\subsection{Direct construction}
One can start with a list of primaries of a certain dimension and attempt
to construct a \WA. One has then to determine the structure constants of the
primaries.
Several methods are in use to check that the OPA is associative.

A first method is based on the claim
\puzzle{bootstrap} in \cite{BPZ} that it is sufficient to check whether all
three- and four-point functions are crossing symmetric. In the
perturbative conformal bootstrap method 
\cite{BPZ,bouw:W24,josestany:sW}, one analyses\col
\be
G^{lk}_{nm}(x) \equiv
\lim_{\stackrel{z\rightarrow \infty}{\e\rightarrow 0}} z^{2h_k}
\langle \Psi_j(z)\Psi_l(1)\Psi_n(x)\Psi_m(\e)\rangle\,.
\ee
The structure constants are then constrained by requiring crossing
symmetry of the $G^{lk}_{nm}(x)$. This can be checked perturbatively
around $x=0$. The crossing symmetry constraints can be implemented using
a group theoretical method due to Bouwknegt \cite{bouw:W24}, that was
generalised in \cite{josestany:sW}.\\
The equations resulting from this method simplify drastically when
considering the $c\rightarrow\infty$ limit.
This is used in \cite{Verstegen:bootstrapinf} to check in a very efficient
way if an algebra exists in this limit, which can be considered to be a
necessary condition for the algebra to exist at generic values of the
central charge $c$ (see also subsection \ref{ssct:BowWatts}). Of course, this
method gives the structure constants only in the large $c$ limit.

Another approach is to check the Jacobi identities with the help of the
mode algebra. In \cite{Bowcock:assoc}, the necessary formulas are given
to compute the commutator of two quasiprimaries in a basis of modes of
quasiprimaries. This was used in 
\cite{blumetal,kauwat,Kausch:WA} 
to explicitly construct the \WA s with a small number
of primary operators of low dimensions.

Finally, one can use OPEs to construct the OPA, and use the formulas of
subsection \ref{ssct:OPErules} (in particular \vgl{eq:OPEJacRAB} for
$q,p>0$) to check the associativity. In
\cite{Klaus:WA5} \OPEdefs\ was used to construct all algebras with
primary generators with dimensions $3,4,5$ and $3,4,5,6$. These are the most
complicated \WA s explicitly constructed up to now.

It is clear that none of these methods can give a classification.  Direct
construction is however the only known way that can give an exhaustive
list of algebras with a certain operator-content.

\subsection{Subalgebras of known \WA s}
Any subalgebra of an OPA trivially satisfies the associativity
conditions. Two main methods exist to construct subalgebras. First, we can
factor out any free fermions and bosons, as is discussed in chapter
\ref{chp:FactFree}. This is a powerful result, as the classification can
now be restricted to \WA s where such operators are not present.

A second method to construct a subalgebra is by ``orbifolding". If a
certain discrete symmetry exists in the \WA, the elements which are
invariant under this symmetry necessarily form a subalgebra. As an
example, the $N=1$ superconformal algebra is invariant under a sign
change of the supersymmetry generator $G$. The corresponding subalgebra
is purely bosonic and is generated by $T$ and two more operators of
dimension $4$ and $6$ respectively \cite{Honeckerextra,Kauschextra}.

Deformable \WA s will contain null operators at a certain value of $c$.  This
gives rise to a quotient algebra.  In many cases, this algebra is only
associative for this value of $c$.  It is a common belief that all
non-deformable algebras can be obtained in this way.  An example of this
mechanism will be given in section \ref{sct:WB2}. Recently, it has been
found that in some cases, a new deformable algebra results
\cite{BONN:unifyWA}. In the quantum case, these new algebras are finitely but
non-freely generated, \ie contain null operators for all values of $c$. In
the classical case, an infinite number of generators is needed to construct
the complete OPA. A particular feature is that the \WA s based on
Lie-algebras in the same Cartan series (see next subsection) seem to contain
the same subalgebra of this new type (for different values of $c$). The study
of these ``unifying" \WA s is based on a formula for the structure constants
of the operators of the lowest dimensions in the \WA\
\cite{Klaus:structureconstants}.

\subsection{Constructing a \WA\ via a
realisation\label{ssct:WAclassrealisation}}
If an algebra is found that is realised in terms of the generators
of an associative algebra, the Jacobi identities are automatically
satisfied. One has to prove that the algebra closes on a finite number of
operators.\\
The main advantage of using a representation is that the
representation theory of the underlying OPA can be used to study the
representations of the \WA\ itself. As we will discuss below,
a whole series of \WA s has been found via realisations in a \KA. Using
the representation theory of the \KA s, the minimal models for the
corresponding \WA s have been completely characterised
\cite{WAreviewpbks}.


A first example is given by the coset construction \cite{GKO:coset,bbss3}.
For a \KA\ $\hat{g}$ of
level $k$ which has a subalgebra $\hat{g}'$, one defines the coset algebra
$\Ww_c[\hat{g}/\hat{g}', k]$ as the set of elements in $\hat{g}$ which
commute with the elements of $\hat{g}'$. The structure of the algebra
generically depends on $k$, due to the appearance of null operators at
specific levels. For certain cases, a representation of the same deformable
algebra is found for all $k$, \eg for a diagonal coset
$\Ww_c[\hat{g}\oplus\hat{g}/\hat{g}_{\sl diag}, k]$.

One can define a second type of coset algebras by considering all elements
in $\hat{g}$ which commute with the originating Lie algebra $g$. These
algebras can be obtained in a certain limit from the diagonal coset algebras
\cite{bbss3}. For a simply-laced Lie algebra $\bar{g}$ and at level $k=1$,
the resulting \WA s are the same as the Casimir algebras defined in
\cite{bbss}. This is proven in \cite{WAreviewpbks} using character
techniques.  For non-simply laced algebras, the situation is more
complicated.  Nevertheless, one still refers to $\Ww_c[\hat{g}/g,k]$ as the
Casimir algebra of $\hat{g}$ at level $k$.\\
As an example, the Casimir
algebra for $B_n$ at level $1$ turns out to be the bosonic projection of
an algebra generated by the Casimir operators of $\hat{B}_n$ and a
fermionic operator of dimension $n+1/2$. For convenience, we call the
underlying \WA, the Casimir algebra of $\hat{B}_n$, $\Ww_c\!B_n$. Section
\ref{sct:WB2} contains an explicit construction of $\Ww_c\!B_2$.

A third method to construct realisations of \WA s is by using
Drinfeld-Sokolov reduction. This method was developed for classical \WA s in
\cite{HRdrisok,bais}, quantisation is discussed in chapter
\ref{chp:WZNW}. Given a (super) \KA, one chooses an
$\slt$ embedding. One then puts certain constraints on the currents of
the \KA. These constraints are characterised by the
$\slt$. As such, one finds a realisation of a finitely generated \WA\
for any $\slt$-embedding of a (super) Lie algebra. The dimensions of the
operators are given by $j_\alpha+1$, where the adjoint representation of the
Lie algebra contains the $\slt$ -irreducible representations
$\underline{j_\alpha}$. Clearly, no dimension $1/2$ operators can be present
in a \WA\ arising from  Drinfeld-Sokolov reduction. This is no severe
restriction, as it is shown in chapter \ref{chp:FactFree} that any \WA\ can
be written as the direct product of some free operators with a \WA\ without
dimension $1/2$ operators.

All known finitely generated \WA s in the class \ref{ass:normalWAs} can be
obtained from Drinfeld-Sokolov reduction, either directly or by applying the
methods of the previous subsection, see \cite{FORT}.  As an example, it is
argued in \cite{WAreviewpbks} that, for simply laced algebras, the diagonal
coset algebras $\Ww_c[\hat{g}\oplus\hat{g}/\hat{g}_{\sl diag},k]$ are the
same as the Drinfeld-Sokolov reductions $\Ww_{DS}\!\![\hat{g},k']$ if one takes
the principal $\slt$ embedding of $g$\footnote{The principal $\slt$ of $g$ is
defined by taking for $e_\pm$ the sum of all positive (resp.\ negative)
simple roots.}. For non-simply laced algebras, the relation between coset and
DS-algebras is more complicated.  As an example, one finds  at least
classically, that $\Ww_c\!B_n =\Ww_{DS}\!\![B(0,n),k]$ 
\cite{ito:WB,wattshr:B(0n)},  for generic $n$, see also section \ref{sct:WB2}.

Before the Drinfeld-Sokolov reduction was quantised, several other
attempts have been made to find a method to construct realisations of
quantum \WA s. We mention only the work of Fateev and Lukyanov
\cite{faly:WAn,faly:WAn2,lufa:HR}, which is based on a quantisation of the
Miura transformation.  This transformation relates different gauge choices
in the constrained phase space, and gives a realisation in terms of
``simpler'' operators \cite{D:thesis}, in particular in terms of free
operators for the simply laced algebras.  In section \ref{sct:WB2}, a free
operator realisation of $\Ww_c\!B_2$ is constructed which is shown to
correspond to the Fateev-Lukyanov construction.

\subsection{Superconformal algebras}
For superconformal algebras, the extra structure given by the
supersymmetry transformations puts strong restrictions on the number of
primary operators in the theory. By studying the Jacobi identities for the
linear superconformal algebras with nonzero central extension, we were
able to classify all possible {\sl linear} algebras with generators of
positive dimension \cite{LinSusy}. The only such algebras that exist have a
number of supersymmetry generators $N\leq 4$. The $N=3$ and the ``large"
$N=4$ algebras appear in section \ref{sct:FFexamples}. The $N=1,2,3$ and the
``small" $N=4$ algebra are all subalgebras of the large $N=4$.
The superconformal algebras with quadratic nonlinearity were classified in
\cite{fradlin}.

\subsection{Attempts towards a classification\label{ssct:BowWatts}}
The methods which rely on a construction of a \WA, either directly or via a
realisation, can give no clue if all \WA s are obtained. In
\cite{Bowcockcontraction}, a first attempt is made to classify all classical
positive-definite \WA s, which are algebras in the class \ref{ass:normalWAs}
with the additional condition that the central extensions define a
positive-definite metric.  It is proven that the finite-dimensional algebra
defined by the linearised commutators of the ``vacuum preserving modes"
$\MO{(\Psi_i)}{m}$ for $|m|<h_i$, is the direct sum of a semisimple Lie
algebra with an abelian algebra\footnote{Free fermions have dimension $1/2$
and hence do not contribute to this Lie algebra.}.  Moreover, this Lie
algebra necessarily contains an $\slt$ formed by the $-1,0,+1$ modes of $T$.
The vacuum preserving modes of $\Psi_i$ form a spin $h_i-1$ representation of
the $\slt$.  This means that by classifying all possible $\slt$-embeddings,
the dimensions which can occur in a classical positive-definite \WA\ are
obtained.  Ref.~\cite{Bowcockcontraction} then proceeds by investigating under
which conditions a quantum \WA\ has a positive-definite
classical limit ($c\rightarrow\infty$).

The algebras arising from Drinfeld-Sokolov reduction (both quantum and
classical) satisfy the criteria of \cite{Bowcockcontraction}, proving
that at least one algebra exists for every $\slt$-embedding. It is not
proven that this algebra is unique. Moreover, for many algebras
constructed by orbifolding no classical limit can be found, and hence
they fall outside this attempt towards classification. For more details,
see \cite{Dublin:vpa,dBT:WAandLA}.

\section{An example : $\Ww_c\!B_2$\label{sct:WB2}}
In this section we will construct $\Ww_c\!\!B_2$, the Casimir algebra of
$\hat{B}_2$. We refer to subsection \ref{ssct:WAclassrealisation} for the
terminology. This algebra contains, besides $T$, an extra dimension $4$
operator and a fermionic dimension $5/2$ operator. The algebra will be
written down in terms of quasiprimary families. We will construct the
most general realisation of this algebra with two free bosons and one
free fermion and show that it is equivalent to the realisation proposed
by Fateev and Lukyanov \cite{lufa:HR}. Finally, using this realisation,
we check that the screening operators are related to the long and short
root of $B_2$ leading to the degenerate representations.

\subsection{The $\Ww_c\!\!B_2$- algebra}
The Lie algebra $B_2$ has two independent Casimir operators of order two and
four. The \emt\ $T$ of $\Ww_c\!\!B_2$ corresponds to the second order
Casimir. The \WA\ contains also a Virasoro primary operator $W$ of dimension
$4$, which corresponds to the fourth order Casimir. Although there exists a
deformable OPA with only this operator content \cite{bouw:W24,hamataka:W24},
this is not yet the Casimir algebra of $B_2$, see subsection
\ref{ssct:WAclassrealisation}. To get $\Ww_c\!B_2$, one has to introduce a
primary weight $5/2$ operator $Q$. This is reminescent of the fact that the
Frenkel-Ka\v{c} level $1$ realisation of the non-simply laced affine Lie
algebras $\widehat{B}_n$, requires, due to the short root, the introduction
of an extra fermion \cite{godetal:FKBn}.

The OPA of $\Ww_c\!\!B_2$ can be written schematically as\col
\bea
Q \times Q &\longrightarrow& \frac{2c}{5} \ [\unity] +
{C_{\frac{5}{2}\frac{5}{2}}}^4  \ [W],\nonumber\\
W\times W &\longrightarrow& \frac{c}{4}[0] + {C_{44}}^4\ [W] + {C_{44}}^6\
[\Psi],\nonumber\\
Q \times W &\longrightarrow& {C_{\frac{5}{2}4}}^\frac{5}{2}\ [Q],
\eea
with $[\unity]$ the conformal family of the identity. From the Jacobi
identities we find the symmetry property ${C_{\frac{5}{2}\frac{5}{2}}}^4
=$ $\frac{8}{5}{C_{\frac{5}{2}4}}^\frac{5}{2} =$ $\frac{8}{5}
{C_{4\frac{5}{2}}}^\frac{5}{2}$. The algebra generated by $\{T, W\}$, for
${C_{44}}^6=0$, is well known to be associative for all values of $c$
\cite{bouw:W24}. Here we discuss the solution where $\Psi$ is the
dimension $6$ Virasoro primary $\Psi \propto P^{-1}(Q, Q)=
\partial Q Q + {\sl corrections}$, see \vgl{eq:Pqdef}. Using Jacobi identities
\re{eq:OPEJacRAB} or the conformal bootstrap, we find that the algebra is
associative for all values of the central charge\footnote{Except, of
course, for these $c$-values for which ${C_{44}}^4$ has poles; the extra
poles in ${C_{44}}^6$ are cancelled by the zeros in the normalisation
constant ${\cal N}$ (\ref{eq:norma}). Notice also that the coupling
constants are imaginary for $-22/5<c<-13/14$ \cite{nahm}.} provided the
couplings are given by\col
\bea
{C_{\frac{5}{2}\frac{5}{2}}}^4  &=&
       \sqrt{\frac{6(14c+13)}{5c+22}}\epsilon_1,\nonumber\\
{C_{44}}^4 &=& \frac{3\sqrt{6} (2c^2+83c-490)}{\sqrt{(14c+13)(5c+22)}
(2c+25)}\epsilon_1,\nonumber\\
{C_{44}}^6 &=&\frac{12\sqrt{5(6c+49)(4c+115)(c-1)}(5c+22)}
{\sqrt{(14c+13)(7c+68)(2c-1)(c+24)}(2c+25)}\epsilon_2,\label{eq:couplings}
\eea
where $\epsilon_1$ and $\epsilon_2$ are arbitrary signs.

For $c=-13/14$ the subalgebra generated by
$\{T, Q\}$ corresponds to the spin 5/2 algebra of Zamolodchikov
\cite{W3zamo}. This is an example of a non-deformable \WA.

As an application of sections
\ref{sct:globalConformalRestrictions} and
\ref{sct:localConformalRestrictions}, we now present the more complicated
OPEs\col
\bea
Q\times Q= \opefive{\frac{2c}{5}}{0}{2T}{\partial T}
{\frac{3}{10}\partial ^2 T +\frac{27}{5c+22}\Lambda
+ {C_{\frac{5}{2}\frac{5}{2}}}^4 W}
\eea
with $\Lambda$ defined in \vgl{eq:Lambdadef}.
For the OPE $W(z)W(w)$, we only list the quasiprimaries appearing in its
singular part.  The OPE can then be reconstructed using eqs.\
(\ref{eq:QPOPE},\ref{eq:QPcoef}).  Denoting $QP^m(W,W)$ as $\phi_{8-m}$, we
have\col
\bea
\phi_0 &=& {c\over 4} \unity,\qquad \phi_2 \ =\ 2T, \nonu
\phi_4 &=& \frac{42}{5c+22}\Lambda\ +\ {C_{44}}^4W,\nonu
\phi_6 &=& -\frac{95c^2 +1254c-10904}{6(7c+68)(5c+22)
    (2c-1)}\left(\partial T \partial T  -\frac{4}{5} \partial ^2TT -
     \frac{1}{42} \partial ^4T\right)\nonu
&&  + \frac{24(72c+13)}{(7c+68)(5c+22)(2c-1)}
      \left((T(TT))-\frac{9}{10}
         \partial ^2TT -\frac{1}{28} \partial ^4T\right)\nonu
&&   -\frac{14}{9(c+24)}{C_{44}}^4\left(\partial ^2W-6 TW\right)\ +\
    {C_{44}}^6\Psi\,.
\eea
Finally, $\Psi$ is the unique dimension $6$ Virasoro primary given by\col
\bea
\Psi &=& {\cal N}\Big( Q\partial Q + \alpha_1 \partial^2 W +
\alpha_2 TW + \alpha_3 \partial T\partial T  +\nonu
&& \alpha_4 T\partial^2 T + \alpha_5 \partial ^4T
+\alpha_6 (T(TT))\Big)\,,
\eea
where the normalisation factor\footnote{This formula corrects a misprint in
\cite{wb2}. ${\cal N}$ is $-5$ times the factor given there.}\col
\be
{\cal N} = -\sqrt{\frac{5(2c-1)(c+24)(7c+68)}{(c-1)(4c+115)(6c+49)(14c+13)}}
\label{eq:norma}
\ee
is fixed such that $\Psi(z)\Psi(w) =(c/6)(z-w)^{-12}+\ldots$
and the coefficients $\alpha_1, \ldots ,\alpha_6$ are\col
\begin{center}
{\small
\begin{tabular}{ll}
$\alpha_1\,=\,-\frac{13c+350}{36(c+24)}{C_{\frac{5}{2}\frac{5}{2}}}^4 ,$&
$\alpha_2\,=\,\frac{19}{3(c+24)}{C_{\frac{5}{2}\frac{5}{2}}}^4,$ \\
$\alpha_3\,=\,- 6(566 c^2+5295c -3998)/N,$&
$\alpha_4\,=\,- 6(530 c^2+6141c -1487)/N,$\\
$\alpha_5\,=\,-( 46 c^3-125 c^2-6235c +1778)/N,$&
$\alpha_6\,=\,12(734c +49)/N,$
\end{tabular}
}
\end{center}
with $N=12(7c+68)(5c+22)(2c-1)$.

The fermionic primary $Q$ can be viewed as some kind of ``generalised
supersymmetry" generator. The appearance of the dimension $4$ operator is
then similar to the appearance of the affine $U(1)$ current in the
$N=2$ super Virasoro algebra. Similarly, one can view the $\Ww_c\!B_n$
algebras as some ``generalised supersymmetry" algebras, the role of the
higher spin operators being to ensure associativity of the OPA for generic
values of the central charge.

\subsection{Coulomb Gas Realisation}
The next step in the analysis of the $\Ww_c\!B_2$ algebra consists in
constructing a Cou\-lomb gas realisation, \ie a realisation of the
primary operators of the OPA in terms of free operators. In order to find this
realisation, one needs two free bosons and one free fermion. The
OPEs for the free operators are defined to be\col
\bea
&\del\varphi _i \times \del \varphi _j  &=\quad \opetwo{\delta_{ij}}{0}\nonu
&\psi \times \psi &=\quad \ope{1}\,.
\eea
The \emt\ is simply the free \emt\ of the two
free bosons, with a background charge, and the free fermion.
Due to rotational invariance, one can always transform the background
charge into one direction. Without limiting the generality, we take \col
\be
T = \frac{1}{2}\partial\varphi _1\partial\varphi_1 +
\frac{1}{2}\partial\varphi _2 \partial\varphi_2
+ \alpha_0 \partial ^2 \varphi_1 + \frac{1}{2} \partial \psi \psi ,
\ee
which satisfies a Virasoro algebra with central charge\col
\be
c=\frac{5}{2} -12 \alpha_0 ^2.
\ee

It is a long and boring task to find the explicit form of the dimension $4$
and dimension $5/2$ primaries.  Using \OPEdefs, one can try to construct the
most general primary dimension four operator and require the $W(z)W(w)$ OPE
to be satisfied. This, however, leads to a system of quadratic equations
which is very difficult to solve. A somewhat easier way is to construct
the most general primary dimension $5/2$ operator. This leads to a four
parameter family of such primaries. The $Q(z)Q(w)$ OPE gives (up to some
discrete automorphisms) three possible solutions for these parameters,
and hence also three candidate dimension four primary operators. Finally,
matching the $W(z)W(w)$ OPE eliminates two of the solutions, and yields a
unique construction. We wish to stress that we have checked all these
statements explicitly using \OPEdefs, including the appearance of the
dimension $6$ primary mentioned earlier.

Let us now present the explicit solution\footnote{By convention, normal
ordering is always from the right to the left, \eg
$\partial \varphi_i\partial \varphi_j\partial \varphi_k =
(\partial \varphi_i(\partial \varphi_j\partial \varphi_k))$.}.
The dimension $5/2$ operator is given by\col
\bea
Q &=& \xi \Big( \frac{3}{2}\partial \varphi_1 \partial \varphi_1 \psi
 - \frac{3}{2}\partial \varphi_2 \partial \varphi_2 \psi +
4\partial \varphi_1 \partial \varphi_2 \psi
+ \alpha_0 \partial ^2 \varphi_1 \psi  + \nonu
&& 3\alpha_0 \partial ^2 \varphi_2 \psi
+ 4 \alpha_0  \partial \varphi_1 \partial\psi
+ 2 \alpha_0  \partial \varphi_2 \partial\psi
+ 2 \alpha_0 ^2 \partial ^2 \psi\Big)\,,
\eea
where $\xi = 1/\sqrt{5(5-4\alpha_0 ^2)}$.
The dimension $4$ operator  is somewhat more complicated\col
\bea
\leqn{W\ =\ \sigma \Big( N^{ijkl} \partial \varphi_i\partial
\varphi_j\partial \varphi_k
\partial \varphi_l + N^{ijk} \partial^2 \varphi_i\partial \varphi_j\partial
\varphi_k+N^{ij}\partial ^3\varphi_i\partial \varphi_j }
&& + \tilde{N}^{ij} \partial^2 \varphi_i\partial ^2\varphi_j  + N^i
\partial^4 \varphi_i + M^{ij}\partial \varphi_i\partial \varphi_j \partial
\psi \psi \nonu
&&+ M^i\partial ^2\varphi_i \partial \psi \psi +\tilde{M}^i
\partial\varphi_i \partial^2 \psi \psi+K_1 \partial ^3 \psi \psi +K_2
\partial ^2 \psi \partial \psi\Big)\,,\label{eq:spin4}
\eea
with $N^{ijkl}$, $\tilde{N^{ij}}$ and $M^{ij}$ completely symmetric and
$N^{ijk}$ symmetric in the last two indices.
The coefficients appearing in (\ref{eq:spin4}) are given explicitly by\col
\begin{center}
{\small
\begin{tabular}{cclccl}
$N^{1111}$&$=$&$N^{2222}= {81}/{80}$,&
$N^{1112}$ &$=$&$-N^{1222}= -{\mu}/{20}$,\\
$N^{1122}$&$=$&$ {(560\alpha_0 ^2-79)}/{720}$, &&&\\
$N^{111}$&$=$&$ {81}\alpha_0/20  $,& $N^{112}$&$=$&$
-{7\alpha_0 }\mu/60$,\\
$N^{122}$&$=$&$ {\alpha_0 (80\alpha_0 ^2+197)}/{60} $,&
$N^{211}$&$=$&$ -{11\alpha_0 }\mu/{30}$,\\
$N^{212}$&$=$&$ -{\alpha_0 }\mu/10 $,&
$N^{222}$&$=$&$ {\alpha_0 }\mu/5$,\\
$N^{11}$&$=$&$ {(128\alpha_0 ^2-25)}/{60} $,&
$N^{12}$&$=$&$ -{\alpha_0 ^2}\mu/{30}$,\\
$N^{21}$&$=$&$ -{\alpha_0 ^2}\mu/5 $,&
$N^{22}$&$=$&$ {(80\alpha_0 ^4+82\alpha_0 ^2-25)}/{60}$,\\
$\tilde{N}^{11}$&$=$&$ {(34\alpha_0 ^2+25)}/{40} $,&
$\tilde{N}^{12}$&$=$&$ -{\alpha_0 ^2}\mu/{15}$,\\
$\tilde{N}^{22}$&$=$&$ {(80\alpha_0 ^4-174\alpha_0 ^2+25)}/{40}$, &&&\\
$N^{1}$&$=$&$ {\alpha_0 (128\alpha_0 ^2-25)}/{360} $,&
$N^{2}$&$=$&$ -{\alpha_0^3 }\mu/30$,\\
$M^{11}$&$=$&$ M^{22}= {(82\alpha_0 ^2-35)}/{6} $,&
$M^{12}$&$=$&$ 0$,\\
$M^{1}$&$=$&$ {\alpha_0 (2\alpha_0 ^2+11)}/{3} $,&
$M^{2}$&$=$&$ {\alpha_0 }\mu/3$,\\
$\tilde{M}^{1}$&$=$&$ -{\alpha_0 }\mu/{3} $,&
$\tilde{M}^{2}$&$=$&$ -{\alpha_0 }\mu/{6}$,\\
$K_{1}$&$=$&$ {(36\alpha_0 ^4-22\alpha_0 ^2+5)}/{18} $,&
$K_{2}$&$=$&$ {(52\alpha_0 ^4-26\alpha_0 ^2-15)}/{6}$,
\end{tabular}
}
\end{center}
where
\be
\sigma  = \sqrt{\frac{3}{2(2-7\alpha_0 ^2)(23-40\alpha_0^2)}}
\frac{1}{4\alpha_0^2-5}, \;\;\;\;\mu=23-40\alpha_0^2.
\label{eq:WB2Wnorm}
\ee
This solution corresponds to the sign choices $\epsilon_1=\epsilon_2=+1$ in
(\ref{eq:couplings}).

Some remarks are in order.

The primary dimension $5/2$ operator $Q$ can be rewritten in a more suggestive
form.  Indeed, rotating the free scalars\col
\bea
\bar{\varphi}_1&=&\frac{1}{\sqrt{10}}(3\varphi_1-\varphi_2),\nonumber\\
\bar{\varphi}_2&=&\frac{1}{\sqrt{10}}(\varphi_1+3\varphi_2),
\eea
one can rewrite\col
\be
Q =5
\left(\sqrt{\frac{2}{5}}\alpha_0 \partial +  \partial \bar{\varphi_1}\right)
\left(\sqrt{\frac{2}{5}}\alpha_0 \partial + \partial
\bar{\varphi_2}\right)\psi ,\label{eq:qfl}
\ee
which is exactly the starting point of the analysis of Fateev and
Lukyanov \cite{lufa:HR}. Our construction hence proves that the algebra
generated by (\ref{eq:qfl}) is indeed finitely generated for all values of the
central charge, as was conjectured in \cite{lufa:HR}.

A second remark concerns the fact that, (up to some discrete
automorphisms), there is (up to some discrete automorphisms) only one
free field realisation of $\Ww_c\!\!B_2$ with two free bosons and one free
fermion. In the case of $\Ww_3$, Fateev and Zamolodchikov found two
inequivalent free field realisations with two free bosons
\cite{fazam:W3freefield}. The simplest one was related to ${su}(3)$ and
led Fateev and Lukyanov to generalise this to the $\Ww\!\!A_n$ algebras
using $n$ free bosons \cite{faly:WAn}. The more complicated one has been
\cite{nar} shown to be related to parafermions (at least for a specific
value of the central charge). In fact, it was argued in \cite{nar} that
such a realisation exists (for fixed $c$) for all $\Ww\!\!A_n$ algebras,
and that in the limit $n\rightarrow \infty$ it corresponds to the $c=2$
free field realisation of $\Ww_\infty$ by Bakas and Kiritsis
\cite{baki:Winf}. For $\Ww_c\!\!B_2$, there does not seem to be a similar
construction\footnote{In \cite{wb2} it was conjectured that no
realisation of $\Ww_{DS}\!\!B_2$ \cite{bouw:W24} exists for generic $c$
using two free scalars. This was based on an explicit calculation with no
background charge $\alpha_0$. However, it was proven in \cite{kauwat:WB2}
that such a realisation does exist, except at isolated values of
$\alpha_0$, including $\alpha_0=0$.}.

\subsection{Highest weight representations}
Examples of $\Ww_c\!\!B_2$-highest weight operators are easily constructed in
the Cou\-lomb gas realisation.  They are given by the vertex operators
defined in subsection \ref{ssct:FreeScalar}\col \be
V_{\vec{\beta}}(z)= e^{\vec{\beta}.\vec{\varphi}}(z)\,,
\label{eq:WB2HWO}
\ee
with $\vec{\beta}\equiv (\beta_1,\beta_2)$ and $\vec{\varphi}(z) \equiv
(\varphi_1(z),\varphi_2(z))$. $V_{\vec{\beta}}(z)$ has Virasoro
dimension\col
\be
\Delta_{\vec{\beta}}=\frac{1}{2}\beta_1^2  +\frac{1}{2}\beta_2^2-\beta_1
\alpha_0,
\ee
To write down the $W$-weight, we introduce the following notation, connected
to the root system of $B_2$.  $\vec{e}_L \equiv \sqrt{\frac{2}{5}}(1,-2)$ and
$\vec{e}_S \equiv \sqrt{\frac{1}{10}}(1,3)$ are the positive simple roots.
$\vec{\rho}= \frac{1}{2} (3\vec{e}_L+4\vec{e}_S)$ is half the sum of the
positive roots; note that $\vec{\alpha}_0 \equiv (\alpha_0 ,0)$ is parallel to
$\vec{\rho}$. Finally, $\eul{W}$ denotes the Weyl group of $B_2$. With this
notation,  the $W$-weight of the vertex operator \re{eq:WB2HWO} can be
written as\col
\bea \leqn{w_{\vec{\beta}} =
\frac{\sigma}{480} \Big(
(40\alpha_0^2-23)\left(\prod_{\eul{w} \in \eul{W}} \eul{
w}(\vec{\beta}-\vec{\alpha}_0).\vec{\rho}\right)^{1/2}}
&& +8(128\alpha_0^2-25)\Delta_{\vec{\beta}} +1944
\Delta_{\vec{\beta}}^2\Big)\label{eq:wwvalue}\,,
\eea
where $\sigma$ was defined in \vgl{eq:WB2Wnorm}.
From this formula, it follows immediately that the weights
$\Delta_{\vec{\beta}}$ and $w_{\vec{\beta}}$ are invariant under
$\vec{\beta} \rightarrow {\eul w}(\vec{\beta}-\vec{\alpha_0 }) +
\vec{\alpha_0 }$, with ${\eul w} \in {\cal W}$ \cite{lufa:HR}.

In the case at hand, one finds at the third order pole $[W\
V_{\vec{\beta}}]_3$ a new Virasoro primary operator, proportional to $ (
\beta_2 \partial \varphi_1 + (2\alpha_0 -\beta_1)\partial \varphi_2))
V_{\vec{\beta}}$. As mentioned in section \ref{sct:WArepresentations}, we
have no intrinsic geometric way to express this new Virasoro primary as well
as the ones appearing in the first and second order poles.

Using the realisation in terms of free fields of the previous section,
one can construct two different kinds of screening
operators. They can be written in such a way that their relation with the
root system of $B_2$ is manifest\col
\bea
V_{\pm}^L &=& \exp (\beta_{\pm}\vec{e}_L.\vec{\varphi})\nonu
V_{\pm}^S &=& \psi\exp (\beta_{\pm}\vec{e}_S.\vec{\varphi})\,,
\eea
with $\beta_+ + \beta_- =\sqrt{\frac{2}{5}} \alpha_0$ and $\beta_+
\beta_- = -1$. They are thus seen to be equal to the screening charges
presented in \cite{lufa:HR}.

Given the screening operators, it is a standard construction to derive the
degenerate representations, see \cite{kama,fazam:W3freefield,lufa:HR,watts}
to which we refer for details.

\section{Discussion}
In this chapter we analysed in detail what the consequences are of the global
and local conformal group on the OPEs in a \WA s. We provided explicit
formulas for working with primaries and quasiprimaries.  Although some of
the formulas are quite complicated, computer implementation presents no
problem \cite{OPEconf}. A further step would be to implement the formulas
of working with OPEs of (quasi)primaries in a future version of \OPEdefs,
\ie without expanding the (quasi)primaries in order to calculate an OPE.

As an example of the power of using the techniques of primary  quasiprimary
operators, combined with automated OPEs, we have proven the existence, for
generic $c$, of the Casimir algebra of $B_2$ by explicitly constructing it.
Using a Coulomb gas realisation in terms of two free bosons and one free
fermion, we have been able to show the equivalence with the results
conjectured by Fateev and Lukyanov \cite{lufa:HR}.

\sectionappendix
\section{Appendix \label{sct:appCommutators}}
This appendix collects some of the more technical details of this chapter.
\subsection*{A few formulas with the modes of the \emt}
First we give a number of identities which follow from the Virasoro algebra
of the modes of the \emt\ \re{eq:VirasoroMode}.

For all $m \in \Bbb{Z}, \, n\in \Bbb{N}$ we have\col
\bea
L_m\, L_{-1}^n &=& \sum_{k=0}^{m+1} \bin{n}{k}
    \Pochhammer{m-k+2}{k}\,L_{-1}{}^{n-k}\, L_{m-k}
    \label{eq:LmL-1n}\\
L_{-1}^n\,L_m  &=& \sum_{k=0}^{m+1} \bin{n}{k}
    \Pochhammer{-m-1}{k} L_{m-k}\,L_{-1}{}^{n-k}
    \label{eq:L-1nLm}\,.
\eea
\subsection*{Derivation of \vgl{eq:Pcomrules}}
We repeat \vgl{eq:Pcomrules} here for convenience\col
\bea
\leqn{\tilde{L}_{m}(h+n)\tilde{L}_{-n}(h)\ =\
   \delta_{m-n} f_1(h,m) + \tilde{L}_{m-n}f_2(h,m,-n)+ }
&&\!\sum_{p\geq 2-\min(n,m)} \tilde{L}_{-n-p}(h\!-\!m\!-\!p)\tilde{L}_{m+p}(h)\
f_3(h,m,-n,p)\,,  \label{eq:coefltmltn}
\eea
where $n,m>1$ for the remainder of this appendix.

We also introduce a shorter notation for the coefficients in
\vgl{eq:Pmdef}\col
\be
\tilde{L}_m(\Phi) = \sum_{n\geq 0} b^m_n(h)
    L_{-1}{}^n\, L_{n+m}\,\Phi\,,
\hspace*{2em}m\in\Bbb{Z}\setminus\{1,0,-1\}\,,
\label{eq:Pmdef1}
\ee
where we defined\col
\be
b^m_n(h)\ \equiv\ \QPs^{m+2}_n(2,h)\ = \ (-1)^n {(2-n-m)_n \over
n!(2h-2m-n-1)_n}\,. \label{eq:Pmcoef1}
\ee
We will need the following lemma.
\begin{lemma}
On a quasiprimary $\Phi$ of dimension $h$, the action of $\tilde L_m$ can
be written as\col
\bea
\tilde{L}_{-m}(\Phi) = \sum_{n\geq 0} \tilde{b}^{m}_n(h)
   L_{n-m} \, L_{-1}{}^n\,\Phi\,,
&&m>1\,, \label{eq:Pmdef2}
\eea
where\col
\be
\tilde{b}^m_n(h)\ =\ (-1)^n \bin{1+m}{n}{
  \Pochhammer{2h + n}{m - n - 2}\over
  \Pochhammer{2h+m+1}{m-2}}\,. \label{eq:Pmcoef2}
\ee
\end{lemma}
The proof follows immediately from the definitions \vgl{eq:Pmdef1} and
\vgl{eq:Pmcoef1} of $\tilde L$ and \vgl{eq:L-1nLm}.

We now set out to prove \vgl{eq:coefltmltn}. Our strategy consists of ordering
all $L_k$ modes in \re{eq:coefltmltn} such that higher modes are moved to the
right. After reordering, we look only at terms which do not contain $L_{-1}$.
This is sufficient as the other terms are fixed by requiring that both \lhs\
and \rhs\ of \vgl{eq:coefltmltn} are quasiprimary.

For the terms in the \rhs\ of \vgl{eq:coefltmltn}, the lemma immediately
gives\col
\bea
\tilde{L}_{-n}(h-m)\tilde{L}_m(h)&\rightarrow&{\Pochhammer{2h
- 2m}{n -2}\over
       \Pochhammer{2h - 2m + n+1}{n-2}}L_{-n}L_m\,,\label{eq:P-nPm}
\eea
where the rightarrow means that we drop terms containing $L_{-1}$.

For the \lhs\ of \vgl{eq:coefltmltn}, we find\col
\bea
\leqn{\tilde{L}_m(h+n)\tilde{L}_{-n}(h)}
&&=\, \sum_{i,j\geq 0}b^m_i(h+n)\,\tilde{b}^{n}_j(h)\,L_{-1}{}^i
\Big( L_{-n+j}L_{m+i}+\nonumber\\
&&\qquad \qquad(m+i+n-j) L_{m+i-n+j}+
        {c\over2}\bin{m+i}{3}\delta_{m+i-n+j}\Big) L_{-1}{}^j\nonu
&&\rightarrow\, \sum_{i,j\geq 0}b^m_i(h+n)\,\tilde{b}^{n}_j(h)\,
\nonumber\\
&&\qquad\Big( \Pochhammer{n\!-\!j\!-\!1}{i}\Pochhammer{m\!+\!i\!-\!j\!+\!2}{j}
       \,L_{-n+j-i}L_{m+i-j}+
   {c\over2}\bin{m}{3} \delta_{m-n}\delta_i\delta_j\nonumber\\
&&\qquad\ + (m\!+\!i\!+\!n\!-\!j)  L_{m-n}\times\nonumber\\
&&\qquad\quad\big(
     \Pochhammer{m\!-\!n\!-\!2}{j} \delta_i\delta _{m-n+j\geq 0} +
     \Pochhammer{n\!-\!m\!-\!i\!-\!1}{i} \delta_j\delta _{m-n+i<0}
    \big) \Big).\label{eq:PmP-n}
\eea
Let us look at the term with $L_{-n+j-i}L_{m+i-j}$.  An additional
reordering of the modes is necessary when $-n+j-i > m+i-j$, which can only
happen if $n>m$. This reordering will give an additional contribution to
the term proportional to $L_{m-n}$ in \vgl{eq:PmP-n}.

We will now find the lower boundary for $p$ in the sum
$\tilde{L}_{-n-p}\tilde{L}_{m+p}$ in \vgl{eq:coefltmltn}. We notice that
the coefficient of $L_{-n+j-i}L_{m+i-j}$ is zero unless $m+i-j+2>0$ or $j=0$.
This means that the lowest $p$ that occurs is certainly larger than $-2-m$.
Furthermore, $p=-1-m$ gives a $L_{-1}$ contribution which we dropped, $p=1-m$
gives $L_1$ which is zero on a quasiprimary and so does not contribute either.
Finally, the term $p=-m$ simply gives $L_0$, and hence $h$, and should be added
to the linear term in \vgl{eq:coefltmltn}. We can conclude that the lowest $p$
which gives a quadratic contribution is larger than or equal to $2-m$, such
that the rightmost $\tilde{L}$ mode of the quadratic term is always a positive
mode. On the other hand, from the definition of $\tilde{b}^{n}_j$
\vgl{eq:Pmcoef2}, we have that $j\leq n+1$. Together with the factor
$\Pochhammer{n\!-\!j\!-\!1}{i}$, this means that for the leftmost
$\tilde{L}$ the highest possible mode has $p=-n$. For the same reasons as
before, we can conclude that the minimal $p$ should also be larger than
or equal to $2-n$ (making the leftmost $\tilde{L}$ always a negative
mode). In this way, we see that \vgl{eq:coefltmltn} has the correct form.

To determine the coefficients in \vgl{eq:coefltmltn}, we compare
\vgl{eq:PmP-n} to \vgl{eq:P-nPm}. Some of the sums in
these coefficients can be found using \Mathematica. We find for $f_1$ and
$f_2$\col  \bea
f_1(h,m)\!&=&\!{c\over 12} (m-1)m(m+1) {\Pochhammer{2h}{m-2} \over
                             \Pochhammer{2h+m+1}{m-2} }\nonu
f_2(h,m,-n)\!&=&\!
  \Bigl((-1)^m (-2\!-\!2 h\!+\!2 h^2\!+\!m\!-\!3 h m\!+\!2 h^2 m\!+\!m^2\!
        -\!h m^2\!-\!n\nonu
&&\qquad +\! 2 h n\!-\!2 m n\!+\!2 h m n\!+\!n^2)
         \Pochhammer{2 \!-\! m \!+\! n}{m} +\nonu
&&\quad\! (2 \!-\! m \!+\! 2 h m \!-\! m^2 \!-\! n \!+\! 2 h n
             \!+\! 2 m n \!+\! n^2)
     \Pochhammer{-2 \!+\! 2 h \!-\! m \!+\! n}{m}\nonu
&&  \Bigr)\Big/\Pochhammer{-2 \!+\! 2 h \!-\! m \!+\! 2 n}{1 + m}\,,
\qquad\qquad\mbox{for $n\geq m$}
\label{eq:Ltf1f2}
\eea
In the case $n<m$, $f_2$ can be determined from the following lemma.
\begin{lemma}
\be
f_2(h,m,-n)= f_2(h-m+n,n,-m)\label{eq:PPf2sym}\,.
\qquad\qquad\mbox{for $n< m$}
\ee
\end{lemma}
\begin{proof}
This relation is most easily proven by considering the inproduct
(for $n>m$)\col
\bea
\leqn{<\Psi|\ \tilde{L}_{n-m}(h)\tilde{L}_m(h+n)\tilde{L}_{-n}(h)\Psi> =}
&& f_2(h,m,-n) <\Psi|\ \tilde{L}_{n-m}(h-m+n)\tilde{L}_{m-n}(h)\Psi>\,,
\label{eq:PPP}
\eea
where $\Psi$ is a primary of dimension $h$ (with $<\Psi|\Psi>$ nonzero) and
we used \vgl{eq:coefltmltn} on the two last operators of the \lhs. The
inproduct is defined in \vgl{eq:modeadjointdef}.
We can compute the inproduct in the \lhs\ of
\vgl{eq:PPP} as\col \be
<\Psi|\ \left(\tilde{L}_{-n}(h)\right)^+\left(\tilde{L}_m(h+n)\right)^+
        \left(\tilde{L}_{n-m}(h)\right)^+\Psi>\,.  \label{eq:3ptlt}
\ee
Now, substituting the definition \vgl{eq:Pmdef} for the rightmost operator,
only the term $L_{m-n}$ of the sum remains, as $L_1\Psi$ is zero. Also,
the first two operators acting on the left state create a quasiprimary
state, which is annihilated (from the right) by $L_{-1}$. Hence, we can
effectively replace $\left(\tilde{L}_{n-m}(h)\right)^+$ by
$\tilde{L}_{m-n}(h)$ in \vgl{eq:3ptlt}. The same reasoning can be
followed for the other operators, but we have to shift the
dimensions\footnote{When computing correlation functions of
quasiprimaries, a similar reasoning shows that
$(\tilde{L}_{n}(h))^+=\tilde{L}_{-n}(h-n)$.}. We get\col
\be
<\Psi|\ \tilde{L}_{n}(h+n)\tilde{L}_{-m}(h-m+n)\tilde{L}_{m-n}(h)\Psi>\,.
\ee
However, using \vgl{eq:coefltmltn} on the two first operators, this is also
equal to\col
\be
f_2(h-m+n,n,-m) <\Psi|\ \tilde{L}_{n-m}(h-m+n)\tilde{L}_{m-n}(h)\Psi>\,,
\ee
which proves \vgl{eq:PPf2sym}.
\end{proof}

To conclude the computation of \vgl{eq:coefltmltn}, we give the
expression for $f_3$\col
\be
f_3(h,n,m,p)= f_4(h,n,m,p){
 \Pochhammer{1\!+\!2h\!-\!2m\!+\!n\!-\!p}{-2+n+p}\over
    \Pochhammer{2 h \!-\! 2 m \!-\! 2 p}{-2+n+p}}\,,
\label{eq:Ltf3}
\ee
where
\bea
\leqn{f_4(h,m,n,p)=}
&&\left\{\begin{array}{ll}
  (-1)^n (1 \!+\! n) \Pochhammer{2 \!+\! m \!-\! n}{n}/
  \Pochhammer{-1 \!+\! 2 h \!+\! n}{n}
    &{\rm if}\ p=-n\\[3mm]
%
  (-2\!+\!m\!+\!m^2\!-\!n\!-\!h n\!+\!2 h^2 n\!-\!m n\!-\!2 h m \\
 \qquad +\! 3 h n^2\!-\!m n^2\!+\!n^3)
   {(-1)^n (1\!+\!n) \Pochhammer{3 \!+\! m \!-\! n}{ -1 \!+\! n}\over
         2 (1 \!-\! h \!+\! m \!-\! n) \Pochhammer{-1 \!+\! 2 h \!+\! n}{n} }
    &{\rm if}\ p =1-n\\[3mm]
%
 (-1)^p\ {}_4{\cal F}_3(
        {-1 \!+\! m, 2 \!+\! m, -1 \!-\! n \!-\! p, 2 \!-\! n \!-\! p};\\
   \qquad\quad
        {2 \!-\! 2 h \!+\! 2 m \!-\! 2 n, 1 \!-\! p, 2 h \!-\! p}; 1
    )\\
 \qquad  {  \Pochhammer{2 h \!-\! p}{1 \!+\! n \!+\! p}
            \Pochhammer{2 \!+\! m \!+\! p}{-p}
            \Pochhammer{2 \!+\! n \!+\! p}{ -p} \over
         (-p)! \Pochhammer{-2 \!+\! 2 h \!+\! n}{1 \!+\! n} }
   & {\rm if}\ 1\!-\!n<p<0\\[3mm]
%
  {}_4{\cal F}_3(
        {-1 \!-\! n, 2 \!-\! n, -1 \!+\! m \!+\! p, 2 \!+\! m \!+\! p};\\
     \qquad\quad
        {2 h, 1 \!+\! p, 2 \!-\! 2 h \!+\! 2 m \!-\! 2 n \!+\! p}; 1
    )\\
   \qquad { \Pochhammer{2 h}{-2 \!+\! n} \Pochhammer{-1 \!+\! m}{p}
            \Pochhammer{-1 \!+\! n}{p} \over
         p! \Pochhammer{1 \!+\! 2 h \!+\! n}{-2 \!+\! n}
       \Pochhammer{-1 \!+\! 2 h \!-\! 2 m \!+\! 2 n \!-\! p}{p} }
   &{\rm if}\ p\geq 0
\end{array}\right.
\eea
We did not find a simple expression for the sums that are involved. We rewrote
them in terms of generalised hypergeometric functions\col
\be
{}_p{\cal F}_q(n_i;m_j;z) =
\sum_{k\geq 0} {\prod_i \Pochhammer{n_i}{k} \over
     k! \prod_j \Pochhammer{m_j}{k}} z^k\,,
\label{eq:HPFQdef}
\ee
where $i$ runs from $1$ to $p$ and $j$ from $1$ to $q$. In $f_4$, the
infinite sum always reduces to a finite number of terms because one of the $n_i$
is negative. One can now use identities for the generalised
hypergeometric functions \cite{Slater} to prove that\col
\be
f_3(h,m,-n,p)=f_3(h+n-m,n,-m,p)\,.\label{eq:PPf3sym}
\ee
Alternatively, this can be checked using an inproduct with four
$\tilde{L}$ operators.

\subsection*{Determination of the coefficient in \vgl{eq:PpartitiononPOPE}}
We first prove\col
\be
\tilde{L}_n(h_j-m) QP^{h_i+m}(\Psi_i,\Psi_j)
= f_5(h_i,h_j,m,n)\ QP^{h_i+m+n}(\Psi_i,\Psi_j)\,,\quad n\geq 2\,.
\label{eq:PnQPm}
\ee
along the same lines as \vgl{eq:coefltmltn}, \ie we
will order the modes, and drop $L_{-1}$ contributions. We write down the
definition of the \lhs\ using modes\col
\be
\sum_{k,l\geq 0} \QPs^{n+2}_k(2,h_j-m) L_{-1}{}^kL_{n+k}\
  \QPs^{m+h_i}_l(h_i,h_j) L_{-1}{}^l \mo{(\Psi_i)}{m+l}\Psi_j\,,
\ee
where the $\QPs^m_n$ are given in \vgl{eq:QPqcoef}. We move the $L_{-1}$ to the
left using \vgl{eq:LmL-1n}, and drop terms containing $L_{-1}$\col
\be
\sum_{l\geq 0} \QPs^{m+h_i}_l(h_i,h_j)\Pochhammer{n-l+2}{l} \
L_{n-l} \mo{\Psi_i}{m+l}\Psi_j\,.
\ee
Now, the factor $\Pochhammer{n-l+2}{l}$ restricts $l$ to be smaller than $n+2$.
In fact, for $l=n+1$ we get a $L_{-1}$ term which we should drop. For $l$ less
than $n$, we can commute $L_{n-l}$ through the $\Psi_i$ mode. Only the
commutator \vgl{eq:PrimaryFieldMode} remains, as $L_{n-l}$ annihilates the
primary operator $\Psi_j$. We get for the \lhs\ of \vgl{eq:PnQPm}, dropping
$L_{-1}$ terms\col
\bea
\leqn{\Big(\sum_{l= 0}^{n-1} \QPs^{m+h_i}_l(h_i,h_j)\Pochhammer{n-l+2}{l}
       (n(h_i-1)-m-l)  }
&&   + \QPs^{m+h_i}_n(h_i,h_j)\Pochhammer{2}{n} (h_j-m-n)\Big)
    \mo{\Psi_i}{m+n}\Psi_j\,.\label{eq:f5def}
\eea
The coefficient of $\mo{\Psi_i}{m+n}\Psi_j$ in this equation is $f_5$, as
the only term in the \rhs\ of \vgl{eq:PnQPm} without $L_{-1}$ is simply
$\mo{\Psi_i}{m+n}\Psi_j$. After summation, we find\col
\bea
\leqn{f_5(h_i,h_j,m,n)=}
&& \Big((-1)^n\big(-h_i(h_i-1) + h_j(h_j-1) + M(M-1) + \nonu
&&\qquad h_j n(2M+n-1)\big) \Pochhammer{h_i - h_j + M}{n} \nonu
&&\ \ +  (h_i(h_i-1) - h_j(h_j-1) + M(M-1) + \nonu
&&\qquad h_i n(2M+n-1)) \Pochhammer{-h_i + h_j + M}{n}\nonu
&&\Big)\Big/\Pochhammer{2M+n-2}{n+1}
\eea
where $M=h_j-m-n$. \\
Although we only looked at the term free of $L_{-1}$, the others have to be
such that the \rhs\ of \vgl{eq:PnQPm} is quasiprimary.  Moreover, we see
that the \lhs\ of \vgl{eq:PnQPm} is of the form $\sum x_l L_{-1}{}^l
\mo{(\Psi_i)}{m+n+l}\Psi_j$. We found in subsection \ref{ssct:QPinOPE} that
requiring this form to be quasiprimary fixed all $x_j$ in terms of $x_0$.

It is now clear that the proportionality constant in
\vgl{eq:PpartitiononPOPE} is given by\col
\be
\tilde{f}(h_i,h_j,h_k,\{\tilde{n}\})\equiv
  \prod_l f_5(h_i,h_j,h_j\!-\!h_k\!-\!\sum_{k\geq l} n_k,\, n_l)\,.
\label{eq:Pftdef}
\ee


\sectionnormal
%12345678901234567890123456789012345678901234567890123456789012345678901234567890
%\mathchardef\Dd="3244
\def\ttt{\tilde{t}}
\def\tg{\tilde{g}}
\def\tu{\tilde{u}}
\def\tq{\tilde{\theta}}
\def\tc{\tilde{c}}
\def\tk{\tilde{k}}
\def\ta{\widetilde{A}}
\def\tG{\widetilde{G}}
\def\tGa{\widetilde{\Gamma}}
\def\tU{\widetilde{U}}
\def\tT{\widetilde{T}}
\def\tW{\widetilde{W}}
\def\tzeta{\tilde{\zeta}}
\def\tgamma{\tilde{\gamma}}
\def\tkappa{\tilde{\kappa}}


\chapter{ Factoring out Free Fields\label{chp:FactFree}}
\mychapter
An Operator Product Algebra (OPA) is factored in two parts if
we can write it as a direct product
structure $\ca \otimes \ca'$. All operators of $\ca$ have nonsingular OPEs
with operators of $\ca'$, or equivalently, their modes commute.
Some years ago, Goddard and Schwimmer \cite{factFermGS} proved that every
OPA can be factorised into a part with only free
fermions (of dimension $\frac{1}{2}$) and a part containing no free fermions.
As a consequence, in the classification of {\WA}s, spin $\frac{1}{2}$
fermions need never be considered. This is very fortunate, since the main
method of constructing a large number of {\WA}s, Drinfeld-Sokolov reduction
(see chapter \ref{chp:renormalisations}), does not yield dimension
$\frac{1}{2}$ fields. (Supersymmetric reduction, see \cite{susyHRFRS},
does give weight $\frac{1}{2}$ fields.)

The first section of this chapter extends the result of Goddard and
Schwimmer for other free fields. We present an algorithmic
procedure for the factorisation. We start with a derivation of the result
of \cite{factFermGS} in our formalism. We then treat bosonic fields of
weight $\frac{1}{2}$, which were not treated in \cite{factFermGS}. It was
already noticed in \cite{factFermGS} that in some cases (\eg the $N=4$
linear superconformal algebra) dimension $1$ bosons can also be decoupled
from a conformal theory. This is certainly not a general property, and
the factorisation-algorithm presented at the end of the first section
gives an easy criterium to decide when free bosons can be decoupled.

In the second section of this chapter, we show how the generating
functionals of the algebra obtained by factoring out free fields
can be found. Also, the criterium for factorisable dimension $1$ bosons
is rederived from Ward identities.

Finally, the linear and nonlinear $N=3,4$ superconformal algebras are
discussed as an example.

The first and second section of this chapter contains material published
in \cite{factfreefield}, see also \cite{D:thesis}. However,
the factorisation algorithm is considerably simplified.
Section \ref{sct:FFexamples} is based on \cite{LinNoLin}.

\section{Algorithms for factorisation\label{sct:FFalgorithms}}
In the following subsections, we will show how various free fields can be
decoupled by introducing certain projection operators on the vectorspace of
fields in the OPA. These operators were found in \cite{factfreefield}, but we
will show some additional properties which make the formulation of the
algorithm simpler.

The method explained in this section is valid in any OPA. In fact, we do
not require the presence of a Virasoro operator. Therefore, we define
the modes $\hat{A}_m$ in this chapter by\col \be
\hat{A}_m B\equiv \left[ AB\right] _m,\hspace*{3em}m\in\Bbb{Z}
\,. \label{eq:FFmodes}
\ee
This is a shift with respect to the usual definition \re{eq:modedef}.

\subsection{Free fermions\label{ssct:FFfermion}}
For completeness, we first rederive the result of \cite{factFermGS} in our
formalism and give an explicit algorithm for the decoupling. Consider a
theory containing a free fermion $\psi$, see section
\ref{ssct:FreeFermion}.
From the OPE \re{eq:psipsiOPE} we find the following anticommutation
relations for the modes\col
\bea
\hat{\psi }_m \hat{\psi }_n = -\hat{\psi }_n \hat{\psi }_m + \lambda \delta
_{m+n-1}\,,&&m,n\in\Bbb{N}\,,
\label{eq:FFcomF}
\eea
where $\lambda$ is a normalisation constant.

Our method consists of defining a set of projection operators $\cp_n$ in
the OPA. $\cp_n$ projects on the kernel of the mode $\hat{\psi}_n$.
Together, they project the OPA to a subalgebra which commutes with $\psi$.
It is then easy to show that the OPA is the direct product of this
subalgebra with the OPA generated by $\psi$.

The projection operators are in the case of free fermions defined by\col
\bea
\cp_n \equiv 1-\frac{1}{\lambda }\hat{\psi }_{1-n}\hat{\psi }_n
\,,&&n>0\,. \label{eq:FFprojnFdef}
\eea
From \vgl{eq:FFcomF} we see that\col
\bea
\hat{\psi}_n\cp_n&=& 0 \nonumber\\
\cp_n\hat{\psi}_{1-n} &=& 0\nonumber\\
\hat{\psi}_n\cp_m&=&\cp_m \hat{\psi}_n,\hspace*{2em}m\neq n,1-n\,.
\label{eq:FFprojF1}
\eea
These relations lead to\col
\bea
\cp_n\cp_n &=& \cp_n\nonumber\\
\cp_n\cp_m&=&\cp_m \cp_n\,.
\label{eq:FFprojF2}
\eea
Together, eqs.\ \re{eq:FFprojF1} and \re{eq:FFprojF2} show that $\cp_n$ is
a projection operator in the kernel of $\hat{\psi}_n$. Moreover, the
different projection operators commute. Clearly, the
projection operator\col
\be
\cp\equiv\prod_{n>0}\cp_n \label{eq:FFprojFdef}
\ee
is such that for any field $A$ of the OPA, the OPE $(\cp A)(z)\ \psi(w)$ is
nonsingular. By using the relation \re{eq:Poleregdef} for the regular part of
an OPE, we see that $\cp A$ is equal to $A$ plus composites containing $\psi$.

As an example, it is easy to check that an \emt\ $T$ for which $\psi$ is
a primary field with dimension $1/2$ gets the expected correction\col
\be
\cp T\ =\ T-\frac{1}{2\lambda }\dz\psi \psi \,,
\ee
\ie the \emt\ of a free fermion is substracted. This means that the central
charge of $\cp T$ is equal to $-1/2$ the central charge of $T$.

When the OPA is generated by operators $T^i$, we see that $\cp T^i$ generate
a subalgebra where all fields commute with $\psi$. Finally, because $\cp
A=A+\ldots$, the complete OPA is generated by $\cp T^i$ and $\psi$. This
proves the factorisation.

\subsection{Symplectic bosons}
Suppose we have a pair of symplectic bosons $\xi^+,\xi^-$ with OPEs given in
section \ref{ssct:bc}. The modes
\re{eq:FFmodes} satisfy the commutation relations\col
\bea
\hat{\xi} ^{\pm}_m\hat{\xi} ^{\pm}_n &=& \hat{\xi} ^{\pm}_n\hat{\xi}
^{\pm}_m,\nonumber\\
\hat{\xi} ^+_m \hat{\xi} ^-_n&=& \hat{\xi} ^-_n\hat{\xi} ^+_m + \lambda
\delta _{m+n-1}\,.
\label{eq:FFcomBC}
\eea
The method we apply is completely similar to the previous case.
We define the operators\col
\bea
\cp ^{\pm}_n \equiv \sum_{i\geq 0}^{} \frac{(\mp1)^{i}}{i!\lambda^i}
\left(\hat{\xi} ^{\mp}_{1-n}\right)^i \left(\hat{\xi}^{\pm}_n\right) ^i\,,
&& n>0\,.
\label{eq:FFprojnBCdef}
\eea
The action of these operators on a field $\Phi$ of the OPA is well-defined when
only a finite number of terms in the sum is non-zero when $\cp^\pm_n$ acts
on $\Phi$. For a graded OPA (see definition \ref{def:dimension}), a
sufficient condition is that ${\rm dim}(\xi^\pm)=1/2$, and that there is a
lower bound on the dimension of the fields in the algebra. In the case
that $h^+={\rm dim}(\xi^+)\geq 1$, a similar argument does not exist.
Indeed, if $n\leq h^+$, it cannot be argued on dimensional grounds that
$\left(\hat{\xi}^+_n\right)^i$ on a field has to be zero for $i$ large
enough, because ${\rm dim}\left(\hat{\xi}^+_n\Phi\right)={\rm
dim}(\Phi)+h^+-n\geq {\rm dim}(\Phi)$. However, we expect that the
projection operators \re{eq:FFprojnBCdef} can be used in most cases.

Assuming that the infinite sums give no problems, we proceed as in the
previous subsection. Using \re{eq:FFcomBC}, we find\col
\bea
\hat{\xi}^\pm_n \cp^\pm_m &=&
   (1-\delta_{m-n})\cp^\pm_m\hat{\xi}^\pm_n\nonu
\cp^\mp_m\hat{\xi}^\pm_n &=&
   (1-\delta_{m+n-1})\hat{\xi}^\pm_n \cp^\mp_m\,.
\eea
This allows us to prove that $\cp^\pm_n$ is a projection operator on the
kernel of $\xi^\pm_n$, and all these projection operators commute. Note
that we can rewrite the definition \re{eq:FFprojnBCdef} as\col
\be
\cp^\pm_n\ =\ :\exp\left(\mp {1\over\lambda}
     \hat{\xi}^\mp_{1-n}\hat{\xi}^\pm_n\right):\,,
\label{eq:FFprojnBCdefexp}
\ee
where normal ordering with respect to the modes is used. The complete
projection operator is\col
\be
\cp\ \equiv\ \prod_{n>0}\cp^+_n\cp^-_n\,.
\ee
This proves that symplectic bosons can all be decoupled when the action of
the operators \re{eq:FFprojnBCdef} is well-defined in the OPA.
This is always the case for dimension $1/2$ symplectic bosons. Note that
when the dimension $h^+\neq 1/2$, fields with zero or negative dimension
are present in the OPA. For applications in conformal field theory this
is undesirable. Of course, $\beta\gamma$--systems used in BRST
quantisation are already factored from the rest of the OPA.

As an example, consider a Virasoro operator $T$. $\xi^+$ and $\xi^-$ are
primaries of dimension $h^+$ and $1-h^+$ respectively with respect to
$T$. We find\col \be
\cp^+_1 \cp^-_1 T = T - {h^+\over \lambda}\xi^+\del \xi^- +
   {h^-\over \lambda}\del\xi^+ \xi^-\,.
\ee
$\cp^+_1 \cp^-_1 T$ already commutes with $\xi^\pm$, so no further
projections are necessary. The central extension of the new Virasoro
operator is given by $c -2(6(h^+)^2 - 6h^+ + 1)$.

Finally, we consider the case of a fermionic $bc$-system. It can be
decoupled using a formula similar to \re{eq:FFprojnBCdefexp}. Here, all
sums are reduce to only two terms, proving that fermionic $b, c$ can
always be factored out. Alternatively, one can define two free fermions
$b\pm c$ for which the results of the previous subsection can be used.
This can be done even if $b$ and $c$ have different dimensions, as
subsection \ref{ssct:FFfermion} does not rely on the dimension at all.

\subsection{$U(1)$ currents}
A (bosonic) $U(1)$ current $J$ has the OPE (the notation for OPEs is
introduced in subsection \ref{ssct:OPAterms})\col
\be
J\times J=\ \opetwo{\lambda}{0}\,.
\ee
The derivative of a free scalar treated in section
\ref{ssct:FreeScalar} provides a realisation of this OPE. The commutation
rules for the modes are\col
\be
\hat{J}_m\hat{J}_n = \hat{J}_n\hat{J}_m+\lambda (m-1)\delta _{m+n-2}\,.
\label{eq:FFU1modes}
\ee
From \vgl{eq:FFU1modes}, we can see that the desired projection operators
are\col
\be
\cp_n\ \equiv\ :\exp\left( {1\over(1-n)\lambda}
     \hat{J}_{2-n}\hat{J}_n\right):\,,
\label{eq:FFprojBdef}
\ee
except for $n = 1$. Notice that in a graded OPA (see def.
\ref{def:dimension}) ${\rm dim}(J)=1$, such that ${\rm
dim}(\hat{J}_1A)={\rm dim}(A)$. Hence, we cannot argue on dimensional
grounds that $(\hat{J}_1)^iA$ is zero even for very large $i$. However,
the situation is disctinctly different from previous subsection, where we
could still hope that for most OPAs the projection operators
\re{eq:FFprojnBCdef} are well-defined. In the case of a $U(1)$-scalar,
$\cp_1$ simply does not exist.

\begin{theorem}
A $U(1)$ current $J$ can be decoupled if and only if\col
\be
\hat{J}_1A = \left[JA\right]_1 =0 \label{eq:FFBcondition}
\ee
for all fields $A$ of the OPA. Or in words, all fields have zero
$\left[U(1)\right]$ charge with respect to the current $J$.
\end{theorem}
\begin{proof}
The fact that \vgl{eq:FFBcondition} is a sufficient condition follows
because we can define the projection operators \re{eq:FFprojBdef}.

We now show that \re{eq:FFBcondition} is also a necessary condition using
a special case of the Jacobi identity \re{eq:OPEJacRAB}\col
\be
[J[BC]_n]_1 = [[JB]_1C]_n + [B[JC]_1]_n \label{eq:FFprojection}
\ee
and
\be
[J\ \del A]_1 = \del([JA]_1)\,.\label{eq:FFJdel}
\ee
Suppose the currents $J$ and $T^k$ generate the algebra and $[JT^k]_1$ is
nonzero for some $k$. We look for an alternative set
$\tilde{T}^k$ which still generate the algebra (together with $J$) and
for which $[J\tilde{T}^k]_1=0$. We can always choose
\be
\tilde{T}^k = T^k + X^k\,,
\ee
where $X^k$ has strictly positive derivative-number (see \vgl{eq:OPEdn}),
or a composite-number (see \vgl{eq:OPEcn}) larger than 1. We define\col
\bea
D = \min_k d([JT^k]_1)&&C = \min_k c([JT^k]_1)\,.
\eea
We will now prove that $d([JX^k]_1)>D$ or $c([JX^k]_1)>C$. If $d(X^k)>0$,
\vgl{eq:FFJdel} proves the first inequality. If $d(X^k)=0$, we
necessarily have that $c(X^k)>1$. In this case the second inequality
follows from \vgl{eq:FFprojection} with $n=0$. This means that it is not
possible to find $X^k$ which cancel the contributions of $T^k$
completely.
\end{proof}

The condition \re{eq:FFBcondition} is not equivalent to $[AJ]_1=0$ because,
see \vgl{eq:OPEcomm}\col
\be
\left[ AJ\right] _1=\left[ JA\right] _1+\sum_{i\geq 2} \frac{(-1)^i}{i!}
\left[ AJ\right] _i\, .\label{eq:FFAJ1}
\ee
In a \WA\ generated by primary fields (except $T$ itself), the criterion becomes
that for any primary generator $A$, $[AJ]_1$ may not contain any primary fields.
Indeed, if $J$ and $C$ are primaries with respect to $T$, we see from
\vgl{eq:FFprojection} with $B=T$ that $[JC]_1$ is primary.  Because the
primary at $[JC]_1$ is the same as the one in $[CJ]_1$,
\vgl{eq:FFBcondition} translates in the requirement that there is no primary
field in $[AJ]_1$.

Finally, we remark that the condition \vgl{eq:FFBcondition} is in fact
quite natural, as $U(1)$-scalars can be viewed as the derivative of a
dimension zero field, and for any fields $A$ and $B$, \vgl{eq:OPEderL}
implies that $[\del A\ B]_1=0$.

\section{Generating functionals}
In this section, we study the relation between the generating functionals
of the algebras related by factoring out a free field. The results are
especially important as these functionals define the induced actions of the
corresponding $\Ww$-gravities (see chapter
\ref{chp:renormalisations}).

Recall the definition of the generating functions \re{eq:Zdef} and
\re{eq:indActdef} for an OPA with generators $T^k$\col
\be
Z[\mu] = \exp\left(-\Gamma [\mu]\right) =
 \left< \exp\left(-\frac{1}{\pi}\int \mu_k T^k\right)\right>\,.
 \label{eq:FFZdef}
\ee

Suppose the OPA contains a free field $F$ that can be factored out. We will
denote by $\tilde{T}^k$ the redefined generators (anti-) commuting with $F$.
By inverting the algorithms of the previous section, we can write\col
\be
T^k = \tilde{T}^k + P^k[\tilde{T},F]\,,\label{eq:FFTTtilde}
\ee
where the $P^k[\tilde{T},F]$ are some differential polynomials with all
terms at least of order $1$ in $F$. We now provide some heuristic
arguments -- \ie based on path integrals -- that the generating
functional $\tilde{Z}[\tilde{\mu}]$ of the reduced gravity theory, which
is defined similarly to \vgl{eq:FFZdef}, can be obtained from $Z[\mu]$ by
integrating over the source of the free field $\mu_F$\col
\be
\tilde{Z}[\tilde{\mu}] =\int [d\mu_{F}]\;Z\left[ \tilde{\mu},\mu_F\right]
\,. \label{eq:FFintegrateF}
\ee
We can compute $Z$ as follows
\be
Z[\tilde{\mu},\mu_F] = \left< \exp\left( -\frac{1}{\pi}\int
\tilde{\mu}_k(\tilde{T}^k + P^k[\tilde{T},F])+\mu_F F\right) \right>_
{\rm OPE}\,. \label{eq:FFZOPE}
\ee
We assume that there exists a path integral formulation for this
expression, \ie the $\tilde{T}^k$ are expressed in terms of some matter
fields $\varphi$. In this case, the polynomials $P^k$ in the path integral
would be the classical limit of those in \re{eq:FFTTtilde}, and some
regularisation procedure has to be applied to find \re{eq:FFZOPE}. In
particular, short distance singularities should be resolved for example
by point splitting. We have\col
\bea
\leqn{Z[\tilde{\mu},\mu_F] = \int [d\varphi][dF]}
\!\!\!&&\!\!\!\exp-\left( S[\varphi]
+S_F[F]+\frac{1}{\pi}\int \tilde{\mu}_k\left(\tilde{T}^k[\varphi] +
P^k[\tilde{T}[\varphi],F]\right)+\mu_F F\right).\label{eq:FFZpathint}
\eea
Here $S_F$ is the free field action which gives the correct OPE for $F$.
For $\mu_k=0$ the matter fields and $F$ are not coupled. This implies that
$\tilde{T}^k$ and $F$ have a non-singular OPE as is required.\\
We now integrate \vgl{eq:FFZpathint} over $\mu_F$ and interchange the order
of integration. The last term in the exponential gives us $\delta(F)$, such
that all terms containing $F$ can be dropped. The remaining expression is
exactly $\tilde{Z}$.

Going to the effective theory (see chapter
\ref{chp:renormalisations}), we define\col
\be
\exp\left(-W[\check{T}]\right) = \int [d\mu]\,
Z[\mu]\exp\left( {1\over\pi} \int \mu_k \check{T}^k\right)\,.
\label{eq:FFdefineW}
\ee
From relation \re{eq:FFintegrateF}, we immediately see\col
\be
\tilde{W}[\check{T}] = W[\check{T}, \check{T}_F =
0]\,.\label{eq:FFWrelation}
\ee
Therefore, the two theories are related by a {\it
quantum} Hamiltonian reduction.

Finally, let us see how the classical limits of the Ward identities for the
generating functionals \re{eq:WIdOPE} are related. \puzzle{}
The first step is to note
that for both theories, the same Ward identities are satisfied by the
Legendre transform\footnote{See the introduction of chapter
\ref{chp:renormalisations} for the definition of these functionals.}
$W^{(0)}$ of the classical limit of $\Gamma$ by
replacing $\{\mu,\delta\Gamma^{(0)}/\delta\mu\}$ with
$\{\delta W^{(0)}/\delta\check{T},\check{T}\}$. Furthermore, $W^{(0)}$ is the
classical limit of $W$ (upto some factors), as it is the saddle-point value in
\re{eq:FFdefineW}. From \vgl{eq:FFWrelation}, we see that the classical
limit of the Ward identities of the reduced theory can be obtained by
putting $\delta\Gamma^{(0)}/\delta\mu_F\,=0$ in the original identities.

Indeed, when we factor out a fermion, the Ward identity corresponding to
$\mu_i = \mu_{\psi }$ is, see \re{eq:WIdOPE}\col
\be
\db u_\psi = -{\lambda\over\pi} \mu_{\psi } + F\left[
\mu, u, u_\psi\right] \,,
\ee
where\col
\be
u^i\equiv \frac{\delta \Gamma }{\delta \mu_i}\,.
\ee
Setting $u_{\psi }=0$, we can solve for $\mu_{\psi}$ and substitute
the solution in the other Ward identities. In this way, the fermion
$\psi $ completely disappears from the theory. The same can be done
for a couple $(\xi ^+,\xi ^-)$ of symplectic bosons, by looking at the
equations with $\mu_i = \xi ^{\pm}$.

We now treat the decoupling of a $U(1)$-scalar $J$. The Ward identity
\re{eq:WIdOPE} of $\mu_J$ has an anomalous term proportional to
$\partial \mu_J$. This means that we will only be able to remove the
scalar field if $\mu_J$ never appears underived. So our criterium for
the factoring out of a scalar field $J$ should be that in all Ward
identities of the theory, the coefficient of $\mu_J$, without
derivative, vanishes. If we look at \vgl{eq:WIdOPE} for some source
$\mu_i$, this term is given by\col
\be
-\frac{1}{\pi}\mu_J\left<\left[ JT _i\right] _1
\exp\left(-\frac{1}{\pi}\int  \mu_k T^k\right)\right>\,.
\ee
We see that requiring this term to vanish, yields precisely the classical
limit of the condition \re{eq:FFBcondition}.

\section{Examples\label{sct:FFexamples}}
In this section we discuss the $N=3,4$ linear
\cite{Susyadem,N4schoutens,N4KUL} and nonlinear
\cite{NLN=3bersh,NLN=3kniz,factFermGS} superconformal algebras.  For both
cases, the factorisations were performed in \cite{factFermGS}.  The relation
\re{eq:FFintegrateF} between the induced actions will be derived here in a
more explicit way without relying on an underlying path integral formalism
for the factorised algebra.
In this section, we do {\em not} follow the conventions of appendix
\ref{app:super} for summation indices. No signs are implied in the
summations and indices are raised and lowered with the Kronecker delta.

\subsection{$N=3$ superconformal algebras}
Both $N=3$ superconformal algebras contain the
\emt\ $T$, supercharges $G^a$, $a\in\{1,2,3\}$ and an
$so(3)$ affine Lie algebra with generators $U^a$, $a\in\{1,2,3\}$. The linear
algebra \cite{Susyadem} contains in addition a dimension $1/2$ fermion $Q$.
The OPEs of the generators are (we use tildes for the nonlinear
algebra and omit OPEs with $T$ and $\tT$)\col
\be
\begin{array}{rcl|rcl}
G^a \,   G^b  &=& \delta^{ab}\frac{2c}{3} [\unity] -\e^{abc} 2 [U^c]
& \tG^a\,  \tG^b&=& \delta^{ab}\frac{2(\tc-1)}{3} [\unity]
                  -\frac{2(\tc-1)}{\tc+1/2}\e^{abc} [\tU^c]\nonu
&&&&&             +\frac 3 {\tc+1/2} [\tU^{(a} \tU^{b)} -
                  { 2\tc+1 \over 3\tc} \delta^{ab} \tT ] \nonu
U^a \,   U^b  &=& -\frac {c}{3} \delta^{ab} [\unity] +\e^{abc} [U^c] &
\tU^a \, \tU^b&=& -\frac {\tc+1/2}{3} \delta^{ab} [\unity] +\e^{abc} [\tU^c] \nonu
U^a \,   G^b  &=& \delta^{ab} [Q]+ \e^{abc}[G^c] &
\tU^a\,  \tG^b&=& \e^{abc} [ \tG^c ] \nonu
Q   \,   G^a  &=& [U^a] &&& \nonu
Q   \,   Q    &=& -\frac c 3[\unity], &&&
\end{array} \label{eq:LNLalgs N=3}
\ee
where we list only the primaries in the OPEs (see section
\ref{sct:localConformalRestrictions}).

The relation \cite{factFermGS} between the linear and nonlinear algebras is
that $Q$ commutes with the combinations that constitute the nonlinear
algebra\col \bea
\tT&\equiv&T+\frac{3}{2c}\del Q Q,
\nonu
\tG^a&\equiv&G^a+\frac 3 c U^aQ,
\nonu
\tU^a&\equiv&U^a,\label{eq:LNLdecoup}
\eea
while the central charges are related by $\tc = c-1/2$. These relations
are easily found by applying the algorithms of section
\ref{sct:FFalgorithms}.

We derive the Ward identities for the induced actions \re{eq:indActdef}
$\Gamma$ and $\tilde\Gamma$ by considering their transformation
properties under $N=3$ supergravity transformations. We will use the
notations
$h\equiv\mu_T,\psi_a\equiv\mu_{G^a},A_a\equiv\mu_{U^a},\eta\equiv\mu_Q$.
The transformations read, for the linear case\col
\bea
\delta h&=&\bdel \e + \e \del h - \del \e h + 2 \theta^a \psi_a,
\nonu
\delta \psi^a&=&\bdel \theta^a + \e \del \psi^a - \frac 1 2 \del \e
\psi^a+\frac 1 2 \theta^a\del h-\del \theta^a h
 -\e^{abc}(\theta_bA_c+\omega_b\psi_c)
\nonu
\delta A^a&=&\bdel \omega^a+\e \del A^a-\e^{abc}
  (\del \theta_b\psi_c-\theta_b\del
\psi_c) +\theta^a\eta-\e^{abc}\omega_bA_c-\del\omega^ah+\tau\psi^a
\nonu
\delta \eta&=&\bdel \tau + \e \del \eta +\frac 1 2  \del \e \eta
+\theta^a\del A_a-\del\omega^a\psi_a
 -\frac 1 2 \tau\del h -\del \tau h\,.\label{eq:LNLsutranslin3}
\eea
They are the same for the nonlinear case, except that there is no
field $\eta$ and no parameter $\tau$, and $\delta A_a$ contains a $\tc$
dependent extra term\col
\be
\delta_{\mbox{\footnotesize extra}}A^a =
 \frac{3}{2\tilde c}\e^{abc}(\del \theta_b\psi_c-\theta_b\del \psi_c)\,.
 \label{eq:LNLdeltaAextra}
\ee
The anomaly for the linear theory is\col
\be
\delta  \Gamma [h,\psi,A,\eta
]=-\frac{c}{12\pi}\int\e\del^3h-\frac{c}{3\pi}\int\theta^a\del^2\psi_a +
\frac{c}{3\pi}\int\omega^a\del A_a+\frac{c}{3\pi}\int\tau\eta\,.
\label{eq:LNLano1} \ee
Defining\footnote{All functional derivatives are left derivatives.}\col
\be
t=\frac{12\pi}{c}\frac{\delta\Gamma }{\delta h} \hspace{1cm}
g^a=\frac{3\pi}{c}\frac{\delta\Gamma }{\delta \psi^a} \hspace{1cm}
u^a=-\frac{3\pi}{c}\frac{\delta\Gamma }{\delta A^a} \hspace{1cm}
q=-\frac{3\pi}{c}\frac{\delta\Gamma }{\delta \eta}
\ee
we obtain the Ward identities for the linear theory by combining eqs.\
\re{eq:LNLsutranslin3} and \re{eq:LNLano1}\col
\bea
\del^3h&=&\nablab t-\left(2\psi_a\del + 6 \del\psi_a\right)g^a
+4 \del A_au^a-\left(2\eta\del-2\del\eta\right)q
\nonu
\del^2 \psi^a&=&\nablab g^a-\frac 1 2 \psi^at+\e^{abc}A_bg^c+\eta u^a
+\e^{abc}\left(2\del\psi_b+\psi_b\del\right)u_c+\del A^a q
\nonu
\del A^a&=&\nablab u^a-\e^{abc}\psi_bg_c+\e^{abc}A_bu_c
-\left(\psi^a\del+\del\psi^a\right) q
\nonu
\eta &=&\nablab q-\psi_au^a\,,\label{eq:LNLWIL3}
\eea
where\col
\be
\nablab\Phi=\left(\bdel-h\del-h_{\Phi}\del h\right)\Phi,
\ee
with $ h_{\Phi}=2,\ \frac 3 2 ,\ 1,\ \frac 1 2$ for $ \Phi=t,\ g^a,\ u^a,\ q$.

Because these functional differential equations have no explicit dependence
on $c$, the induced action can be written as\col
\be
\Gamma [h,\psi,A,\eta ]=c\ \Gamma^{(0)} [h,\psi,A,\eta ]\,,
\label{eq:LNLohwell}
\ee
where $\Gamma^{(0)}$ is $c$-independent.

The nonlinear theory can be treated in a parallel way. The anomaly is
now\col \bea
\delta  \tGa [h,\psi,A ]&=&-\frac{\tc} {12\pi}\int\e\del^3h-
\frac{\tc-1}{3\pi}\int\theta^a\del^2\psi_a +
\frac{\tc+1/2}{3\pi}\int\omega^a\del A_a\nonu
&&-\frac{3}{\pi (\tc+1/2)}\int\theta_a\psi_b
   \left(U^{(a}U^{b)}\right)_{\mbox{eff}}\,.
\label{eq:LNLanomNL2pre}
\eea
The last term, which is due to the nonlinear term in the algebra
\vgl{eq:LNLalgs N=3}, can be rewritten as\col
\bea
\left(U^{(a}U^{b)}\right)_{\mbox{eff}}(x)\!\!&=&\!\!
\Big\langle \,\tU^{(a} \tU^{b)}(x) \exp\Big(-\frac{1}{\pi} \int
 \Bigl( h \tT + \psi_a \tG^a + A_a \tU^a\Bigr)\Big)\Big\rangle\Big/\nonu
&&\exp \Big(-\tGa\Big) \label{eq:LNLeffreg}\\
&=&\!\!\left(\frac {\tc+1/2}{3} \right)^2 u^a(x)\,u^b(x)+\nonu
&&\frac{(\tc+1/2)\pi}{6}
\lim_{y\rightarrow x}\biggl(\frac{\del u^a (x)}{\del A_b(y)}
 -\frac{\del}{\bdel}\delta^{(2)}(x-y)\delta^{ab} + a\rightleftharpoons
b \biggr)\,. \nonumber
\eea
The limit in the last term of \vgl{eq:LNLeffreg} reflects the
point-splitting regularisation of the composite terms in the $\tG \tG$
OPE \re{eq:LNLalgs N=3}.  One notices that in the limit
$\tc\rightarrow\infty$, $u$ becomes $\tc$ independent and one has simply\col
\be
\lim_{\tc\rightarrow\infty}
     \left( \frac {3}{\tc+1/2} \right)^2
       \left(U^{(a}U^{b)}\right)_{\mbox{eff}}(x)= u^a(x) \, u^b(x)\,.
\label{eq:LNLeffreglc}
\ee
Using \vgl{eq:LNLeffreg}, we find that \vgl{eq:LNLanomNL2pre} can be
rewritten as\col \bea
\delta  \tGa [h,\psi,A ]&=&-\frac{\tc} {12\pi}\int\e\del^3h-
     \frac{\tc-1}{3\pi}\int\theta^a\del^2\psi_a +
     \frac{\tc+1/2}{3\pi}\int\omega^a\del A_a\nonu
&&   -\frac{ \tc+1/2}{3\pi}\int\theta_a\psi_b u^a u^b \nonu
&&-\lim_{y\rightarrow x} \int \theta^{\,(a}\psi^{b)} \left(\frac{\del u^a (x)}{\del
A_b(y)}-\frac{\del}{\bdel}\delta^{(2)}(x-y)\delta_{ab}\right)\,,
 \label{eq:LNLanomNL3}
\eea
where the last term disappears in the large $\tc$ limit.  The term
proportional to $\int\theta_a\psi_b u^a u^b$ in \vgl{eq:LNLanomNL3} can be
absorbed by adding a field dependent term in the transformation rule for
$A$\col
\be
\delta^{\rm nl}_{\mbox{\footnotesize extra}}A_a=-\theta_{a}\psi_{b}u^b\,.
\ee
Doing this, we find that in the large $\tc$ limit, the anomaly reduces to
the minimal one.

Combining the nonlinear transformations with \vgl{eq:LNLanomNL3},
and defining\col
\be
\ttt=\frac{12\pi}{\tc}\frac{\delta\tGa}{\delta h}\hspace{1cm}
\tg^a=\frac{3\pi}{\tc-1}\frac{\delta\tGa}{\delta \psi_a}\hspace{1cm}
\tu^a=-\frac{3\pi}{\tc+1/2}\frac{\delta\tGa}{\delta A_a},
\label{eq:LNLdefcurr}
\ee
we find the Ward identities for $\tGa [h,\psi,A ]$ (they can also
be found in \cite{ZFacN=3})\col
\bea
\del^3h&=&\nablab\ttt-\left(1-\frac{1}{\tc}\right)\left(2\psi_a\del +
        6 \del\psi^a\right)\tg^a
+4 \left( 1+\frac{1}{2\tc}\right)\del A_a \tu^a
\nonu
\del^2 \psi_a&=&\nablab\tg^a-\left(\frac 1 2 + \frac{1}{2\tc-2}\right)
\psi_a\ttt+\e^{abc}A_b\tg^c
+ \e^{abc} \left(2\del\psi_b+\psi_b\del\right) \tu^c\nonu
&&- \left(1+\frac{3}{2\tc-2}\right)\left(3\over \tc+1/2\right)^2
\psi_b \left(U^{(a}U^{b)}\right)_{\mbox{eff}}
\nonu
\del A_a&=&\nablab\tu^a-\left(1-\frac{3}{2\tc+1}\right)
\e^{abc}\psi_b\tg^c+\e^{abc}A_b\tu^c\,.\label{eq:LNLWINL3}
\eea

The normalisation of the currents has been chosen such that the anomalous
terms on the \lhs\ have coefficient one. The explicit $\tc$ dependence of
the Ward identities arises from several sources\col\ some couplings in the
nonlinear algebra \vgl{eq:LNLalgs N=3} are explicitly
$\tc$-dependent, the transformation rules \vgl{eq:LNLdeltaAextra} are
$\tc$-dependent, and the field-non\-linear\-ity. The dependence implies
that the induced action is given by a $1/\tc$ expansion\col
\be
\tGa[h,\psi,A]=
 \sum_{i\geq 0}\tc^{1-i}\tGa^{(i)}[h,\psi,A] \ {}.
\ee

Returning to the Ward identities for the linear theory \vgl{eq:LNLWIL3},
we observe that when we take $\tilde{c}=c+1/2$ and put $q=0$, we find from
the last identity in \vgl{eq:LNLWIL3} that $\eta =-\psi_au^a$.  Substituting
this into the first three identities of \vgl{eq:LNLWIL3} yields
precisely the Ward identities for the nonlinear theory \vgl{eq:LNLWINL3} in
the $c\rightarrow\infty$ limit.  Also, the extra term in the nonlinear $\delta
A_a$ (\vgl{eq:LNLdeltaAextra}), that was added to bring the anomaly to a
minimal form, now effectively reinserts the $\theta^a \eta$ term that
disappeared from the linear transformation, \vgl{eq:LNLsutranslin3}. This is
in accordance with the observations at the end of the previous section.

We will now prove relation \re{eq:FFintegrateF} between $Z$ and $\tilde
Z$. First we rewrite the definition of $Z$ \re{eq:Zdef} using
\vgl{eq:LNLdecoup}, the crucial ingredient being that $Q$ commutes with the
nonlinear algebra, thus factorising the averages\col \bea
Z [h,\psi,A,\eta ]
\!\!&=&\!\!\bigg\langle\!\!\exp\Big( -\frac{1}{\pi}
\int ( h \tT + \psi_a \tG^a + A_a \tU^a )\Big)\nonu
&&\!\!\hspace*{2em}\Bigl\langle
    \exp\Big( -\frac{1}{\pi} \int ( h T_Q + \hat{\eta}Q)\Big)
\Bigr\rangle_{Q}\,\biggr\rangle\nonu
&=&\!\!\bigg\langle\!\!\exp\Big( -\frac{1}{\pi} \int ( h \tT + \psi_a \tG^a
+ A_a \tU^a )-\Gamma [h,\hat{\eta}]\Big)\!\! \biggr\rangle
\label{eq:LNLqdec}
\eea
where\col
\bea
&T_Q=\frac{3}{2c} Q\del Q,& \hat{\eta}=\eta-\frac{1}{3c} \psi_a\tU^a\,.
\eea
The $Q$ integral can be expressed in terms of the Polyakov action
\re{eq:explicitGammaT}\col
\be
\Gamma [h,\hat{\eta}] = \frac{1}{48\pi} \Gamma_{\mbox{Pol}}[h] - \frac{c}{6\pi}
\int\hat{\eta} \frac{1}{\nablab}\hat{\eta}\,,\label{eq:LNLfermion}
\ee
where   $ \nablab=\delb-h\del-\frac 1 2 \del h $ and
\be
\Gamma_{\mbox{Pol}} [h]=\int\del^2h \frac{1}{\bdel}
\frac{1}{1-h\del\delb^{-1}} \frac{1}{\del}\del^2h\,.
\ee
Using eqs.\ \re{eq:LNLqdec} and \re{eq:LNLfermion}, we find\col
\bea
\lefteqn{\exp\Big( -\tGa[h,\psi,A]\Big)\ =}\\ \label{eq:LNLnll2}
%\exp\Big[ \frac{1}{48\pi}\Gamma_{\mbox{Pol}}[h] \Big]
%\exp\Big[ -\frac{c}{6\pi}\int\Bigl(\eta+\frac{\pi}{3c}\psi_a\frac{\delta}{\delta
%A_a}\Bigr)\frac{1}{\nablab}
%\Bigl(\eta+\frac{\pi}{3c}\psi_b\frac{\delta}{\delta A_b}\Bigr) \Big] \nonu
%&&\qquad\exp \Big[ -\Gamma [h,\psi,A,\eta ] \Big].\label{eq:LNLnll2}
&&\exp\Big(\Gamma [h,
    \hat{\eta}= \eta+\frac{\pi}{3c}\psi_b\frac{\delta}{\delta A_b}]\Big)
    \,\exp \Big( -\Gamma [h,\psi,A,\eta ] \Big)\,.\nonumber
\eea
We checked this formula explicitly on the lowest order correlation
functions using \OPEdefs.
Introducing the Fourier transform of $\Gamma$ with respect to\ $A$\col
\be
\exp\Big( -\Gamma [h,\psi,A,\eta ]\Big) =\int [du]\exp\Big( -\Gamma
[h,\psi,u,\eta ]+\frac{c}{3\pi}\int u^aA_a\Big)\,, \ee
\vgl{eq:LNLnll2} further reduces to\col
\bea
\exp\Big(-\tGa[h,\psi,A] \Big)\!\!\!&=&\!\!\!
\exp \Big( \frac{1}{48\pi}\Gamma_{\mbox{Pol}}[h] \Big)
\int [du] \exp\Big( -\Gamma [h,\psi,u,\eta ] \\ \label{eq:LNLnll3}
&&-\frac{c}{6\pi}\int\Bigl(\eta+\psi_au^a\Bigr) \frac{1}{\nablab}
\Bigl(\eta+ \psi_bu^b\Bigr)+\frac{c}{3\pi}\int u^aA_a\Big)  \,.\nonumber
\eea
As the \lhs\ of \vgl{eq:LNLnll3} is $\eta$-independent, the \rhs\ should
also be. We can integrate both sides over $\eta$ with a measure chosen such
that the integral is equal to one\col
\be
\exp\Big(   -\frac{1}{48\pi}\Gamma_{\mbox{Pol}}[h]\Big)
\int [d\eta] \exp\bigg( \frac{c}{6\pi}\int\Bigl(\eta+\psi_au^a\Bigr)
\frac{1}{\nablab}
\Bigl(\eta+ \psi_bu^b\Bigr)\bigg) =1\,.
\ee
Combining this with \vgl{eq:LNLnll3}, we finally obtain
\vgl{eq:FFintegrateF}.

\subsection{$N=4$ superconformal algebras\label{ssct:FFN=4}}
Now we extend the method applied for $N=3$ to the case of
$N=4$. Again, there is a linear $N=4$ algebra and a
nonlinear one, obtained \cite{factFermGS} by
decoupling four free fermions and a $U(1)$ current.
In the previous case we made use in the derivation of the
explicit form of the action induced by integrating out the fermions. In the
present case no explicit expression is available for the corresponding
quantity, but we will see that it is not needed.

The $N=4$ superconformal algebra \cite{N4schoutens,N4KUL} is generated by the
\emt\ $T$, four supercharges $G^a$, $a \in {\{}1,2,3,4{\}}$, an $so(4)$ affine
Lie algebra with generators $U^{ab} = -U^{ba}$, $a,b \in {\{}1,2,3,4{\}}$, 4
free fermions $Q^a$ and a $U(1)$ current $P$. The two $su(2)$- algebras  have
levels $k_+$ and $k_-$. The supercharges $G^a$ and the dimension $1/2$ fields
$Q^a$ form two $(2,2)$ representations of $su(2) \otimes su(2)$. The central
charge is given by\col
\be
c = {{6\,k_{+}\,k_{-}}\over {k_{+} + k_{-}}}\, .
\ee
The OPEs are (we omit the OPEs of $T$)\col
\bea
G^{a}\,G^{b} &=&
  {3c\over2} \delta ^{ab}[\unity] +
  [ -2\,U^{ab} + \zeta \,\e ^{abcd} U^{cd}]
\nonu
U^{ab}\,U^{cd} &=&
  {k\over 2} \left( \delta^{ad}\,\delta ^{bc} - \delta^{ac}\,\delta ^{bd} -
                    \zeta \,\e ^{abcd}
             \right) \,   [\unity] \nonu
&& +  [{\delta ^{bd}\,U^{ac} - \delta ^{bc}\,U^{ad} -
               \delta ^{ad}\,U^{bc} + \delta ^{ac}\,U^{bd}]}
\nonu
U^{ab}\,G^{c} &=&
       - \zeta \, \left( \delta ^{bc} \,[Q^{a}] - \delta ^{ac}\,[Q^{b}]
                  \right) + \e ^{abcd}\,[Q^{d}]
 - (\delta ^{bc}\,[G^{a}] - \delta ^{ac}\,[G^{b}])
\nonu
Q^{a}\,G^{b} &=&
   \delta ^{ab}\,[P] - {1\over 2}\e ^{abcd}\,[U^{cd}]
\nonu
Q^{a}\,U^{bc} &=&
  \delta ^{ac}\,[Q^{b}] - \delta ^{ab}\,[Q^{c}]
\nonu
P\,G^{a} &=& [Q^{a}]
\nonu
P\,P &=& -{k\over2}[\unity]
\nonu
Q^{a}\,Q^{b} &=&
  - {k\over 2}\delta ^{ab}[\unity]
   \label{eq:LNLlinalg}
\eea
where $k = k_{+} + k_{-}$ and $ \zeta= (k_{+} - k_{-})/ k$. Note that
$[PA]_1=0$ for all primaries, and hence for all elements of the OPA.


We will write the induced action \re{eq:indActdef} as $\Gamma[h, \psi, A,
b,\eta]$.  All the structure constants of the linear algebra
(\ref{eq:LNLlinalg}) depend only on the ratio $k_+/k_-$.  Apart from this
ratio, $k$ enters as a proportionality constant for {\it all} two-point
functions.  As a consequence, $\Gamma$ depends on that ratio in a nontrivial
way, but its $k$-dependence is simply an overall factor $k$.

Using the following definitions\col
\be
  t     = {12\pi \over c}  \ddt{\Gamma}{h}, \hspace{1em}
  g^a   = {3\pi \over c}   \ddt{\Gamma}{\psi_a}, \hspace{1em}
  u^{ab}= -{\pi \over k}   \ddt{\Gamma}{A_{ab}}, \hspace{1em}\nonumber
\ee
\be
  q^a   =-{2\pi \over k}  \ddt{\Gamma}{\eta_a},  \hspace{1em}
  p     =-{2\pi \over k}  \ddt{\Gamma}{b}
\ee
and $ \gamma  = 6k / c$,
the Ward identities are\col
\bea
\del^3 h &=& \nablab t
   - 2\left(\psi _a  \del + 3 \del \psi _a \right ) g^a
   + 2\,\gamma \,\del A_{ab}  u^{ab}
   +{\gamma \over 2}\,\left(\del \eta _a- \eta _a  \del \right) q^a
   + \gamma \,\del b \, p
   \nonu
\del^2 \psi _a &=& \nablab g^a
   -2\,A_{ab}  g^b
   - {1\over 2}\psi _a  t
   + {\gamma\over 4} \del b \, q^a
   + {\gamma\over 2} \zeta \,\del A_{ab}  q^b
   + {\gamma\over 4} \e _{abcd}\,\del A_{bc}  q^d
   \nonu
&& + {\gamma\over 4}\e _{abcd}\,\eta _b  u^{cd}
   + {\gamma \over 4} \left(\psi _b \del + 2 \del \psi _b\right)
       (2u^{ab} - \zeta \e _{abcd}\, u^{cd})
   + {\gamma\over 4} \,\eta _a  p
   \nonu
\del A_{ab} &+& {{\zeta}\over 2} \e _{abcd}\,\del A_{cd} =
\nablab u^{ab}
   -4\,A_{c[a}  u^{b]c}
   - {4\over {\gamma }}\psi _{[a}  g^{b]}
   + \eta _{[a}  q^{b]}
   \nonu
&& - \zeta \left( \psi _{[a} \del + \del \psi_{[a} \right) q^{b]}
   - {1\over 2}\e _{abcd}\,\left( \psi _c \del + \del \psi_c \right) q^d
   \nonu
\eta_a &=& \nablab q^a
   -2\,A_{ab}  q^b - \psi _a  p +
   \e _{abcd}\,\psi _b  u^{cd} \nonu
\del b &=& \nablab p
   -\left( \psi _a  \del + \del \psi _a \right) q^a \, .
   \label{eq:LNLlinearWard4}
\eea
Note that $b$ appears only with at least one derivative. The brackets
denote antisymmetrisation in the indices.

The nonlinear $N=4$ superconformal algebra has the same structure as
\vgl{eq:LNLlinalg} but there is no $P$ and $Q^a$. The central charge is
related to the $su(2)$-levels by $\tc={3 (\tk + 2\tk_+ \tk_-) \over
2 + \tk}$.  We only give the $\tG \tG$ OPE explicitly\col
\begin{eqnarray}
\tG^{a}\,\tG^{b} &=&
  {{4 \tk_{+} \tk_{-} }\over \tk+2} \, \delta ^{ab} \,[\unity]
   -  {2\tk \over \tk+2}  \, [\tU^{ab}]  +
       {\tk_+ - \tk_-\over \tk+2} \e _{abcd}\,[\tU^{cd}]
    \\
&& +
[{2\tk \over \tk + 2 \tk_+ \tk_-} \,\delta ^{ab}\,\tT +
       {1 \over 4(\tk+2)}\,\e _{acdg}\,\e _{befg}
      (\tU^{cd} \tU^{ef} +  \tU^{ef} \tU^{cd})]\,.\nonumber
%\nonu
%\tU^{ab}\,\tU^{cd} &=&
% {1\over 2}\left( \tk \,
%            ( \delta^{ac}\,\delta ^{bd} - \delta ^{ad}\, \delta ^{bc}) -
%      {( \tk_{+} - \tk_{-}) \e _{abcd}}
%     \right) [\unity] \nonu
%&&+
%  [\delta ^{bd}\,\tU^{ac} -
%      \delta ^{bc}\,\tU^{ad} -
%      \delta ^{ad}\,\tU^{bc} +
%      \delta ^{ac}\,\tU^{bd}]
%\nonu
%\tU^{ab}\,\tG^{c} &=&
%   - \delta ^{bc}\,[\tG^{a}] + \delta ^{ac}\,[\tG^{b} ]
\end{eqnarray}

To write down the Ward identities in this case, we define\col
\be
  \ttt    =  {12\pi \over \tc}       \ddt{\tGa}{h}, \hspace{1cm}
  \tg^a   = {(\tk+2) \pi\over 2\tk_+\tk_-} \ddt{\tGa}{\psi_a}, \hspace{1cm}
  \tu^{ab}= -{\pi\over \tk}          \ddt{\tGa}{A_{ab}},
\ee
and
\be
  \tgamma = {\tk (\tk+2) \over \tk_{+} \tk_{-}}, \hspace{1cm}
  \tkappa = {6 \tk \over \tc},\hspace{1cm}
  \tzeta = {\tk_{+}-\tk_{-} \over \tk}\, .
\ee
\bea
 \del^3 h&=&\nablab \ttt
   -{2\tkappa\over \tgamma} \left( \psi_a \del + 3 \del \psi_a\right) \tg^a
   + 2 \tkappa\,\del A_{ab} \tu^{ab}  \nonu
\del^2 \psi _a &=& \nablab \tg^a
   -2\,A_{ab}  \tg^b
   -{\tgamma\over 2\tkappa}\,\psi _a \ttt\nonu
&& -{\tgamma\over 4 \tk (\tk+2)}
       \,\e _{acdg}\, \e _{befg}
       \, \psi _b \left(
           \left( \tU^{cd} \tU^{ef} \right)_{\it eff} +
           \left( \tU^{ef} \tU^{cd} \right)_{\it eff}
       \right)
       \nonu
&& + {\tgamma \tk \over 4(\tk+2)}\,(\psi_b \del +2\del \psi_b)
      ( 2\tu^{ab} - \tzeta \, \e _{abcd} \tu^{cd} )
   \nonu
\del A_{ab} &+& {\tzeta\over 2}\,\e _{abcd} \, \del A_{cd} =
\nablab \tu^{ab}
   -4 A_{c[a}  \tu^{b]c}
   -{4\over \tgamma} \psi _{[a}  \tg^{b]}\,.
\eea

As in the previous subsection, we will use the results of
section \ref{sct:FFalgorithms} or \cite{factFermGS} to eliminate the free
fermion fields and the $U(1)$-field $P$.  The new currents are\col
\bea
\tT &=& T + {1\over k} P P +
   {1\over k}\del Q^{c} Q^{c}\nonu
\tG^{a} &=& G^{a} + {{2\over k}P Q^{a}} +
   \e _{abcd}\,
    \left( {2\over 3k^2}\,Q^{b} Q^{c} Q^{d} +
      {1\over k}{Q^{b} \tU^{cd}} \right) \nonu
\tU^{ab} &=& U^{ab} - {2\over k}\,Q^{a} Q^{b}
 \label{eq:LNLdecompo}
\eea
and the constants in the algebras are related by
$\tk_\pm = k_\pm - 1$, and thus $ \tc = c-3$.
Again, we find agreement between the large $k$-limit of the Ward identities
putting $q^a$ and $p$ to zero, and solving $\eta^a$ and $\partial b$ from the
two last identities of (\ref{eq:LNLlinearWard4}).

We now find the nonlinear effective action in terms of the
effective action of the linear theory. In analogy with the $N=3$ case, it
seems that the operators of the nonlinear theory can be written
as the difference of the operators of the linear theory, and a realisation of
the linear theory given by the free fermions and $P$. In the present case,
this simple linear combination is valid for the integer spin currents
$\tT$ and $\tU$, but not for $\tG$. A second complication is that, due to
the presence of a trilinear term (in $Q$) in the relation between $\tG$
and $G$, integrating out the $Q$-fields is more involved. Nevertheless,
we can still obtain \re{eq:FFintegrateF}.

There is a variety of ways to derive this relation, starting by
rewriting the decompositions of (\ref{eq:LNLdecompo}) in different ways.
We will use the following form\col
\begin{equation}
G^a + {1 \over k} \e _{abcd} Q^b U^{cd} =
\tG^a - {2\over k} P Q^a + {4 \over {3 k^2}} \e _{abcd} Q^b Q^c Q^d
\, .
\end{equation}
This leads immediately to\footnote{%
There are no normal ordering problems as the OPEs of the relevant
operators turn out to be nonsingular (\eg the term cubic in the $Q^a$
is an antisymmetric combination).}\col
\bea
\lefteqn{
  \Big\langle\exp\bigg(
    -\frac{1}{\pi}  \int
         \Bigl( h T+\psi_a G^a + A_{ab} U^{ab}  + b P+ \eta_a  Q^a
             + {1\over k} \e_{abcd} \psi_a Q^b U^{cd}
         \Bigr)
   \bigg)  \Big\rangle \, =
} \nonu
&&  \Big\langle\exp\bigg( -\frac{1}{\pi} \int
      \Bigl(
      \left(h \tT+\psi_a \tG^a+ A_{ab} \tU^{ab} \right) \label{eq:LNLstartind4}\\
 &&\ \ +{1\over k}\left(-h P^2-h \del Q^a Q_a- 2 \psi_a P Q^a
           + 2 A_{ab} Q^a Q^b + b P + \eta_a  Q^a \right)\nonu
 &&\ \ + {4 \over {3 k^2}} \e _{abcd}\psi_a Q^b Q^c Q^d
\Bigr)\bigg) \Big\rangle\, . \nonumber
\eea
Again the crucial step is that in the \rhs, the expectation value
factorises\col\ the average over $Q^a$ and $P$ can be computed separately,
since these fields commute with the nonlinear SUSY-algebra.
This average is in fact closely related to the partition function
for the linear $N=4$ algebra with $k_+=k_-=1$ and $c=3$, up to the
renormalisation of some coefficients. We have\col
\bea
\lefteqn{Z^{c=3}[h,\psi,A,b,\eta]\ =\ \Big\langle\exp\bigg( -\frac{1}{\pi} \int
       \Bigl(-{h\over 2} \left(\hat{P}\hat{P} +\del \hat{Q}_a \hat{Q}^a
       \right)}\label{eq:LNLZetdrie}\\
&&\quad -\psi_a \left(\hat{P}\hat{Q}^a+{1 \over 6}\e _{abcd}
  \hat{Q}^b\hat{Q}^c\hat{Q}^d \right)
 +A_{ab}\hat{Q}^a \hat{Q}^b+b \hat{P} +\eta_a \hat{Q}^a
\Bigr)\bigg) \Big\rangle \nonumber
\eea
where the average value is over free fermions $\hat{Q}^a$ and a free
$U(1)$-current $\hat{P}$. These are normalised in a $k$-independent
fashion\col \bea
\hat{P}\times\hat{P}\rightarrow-[\unity]&&
\hat{Q}^a\times\hat{Q}^b\rightarrow -\delta ^{ab}[\unity]
\eea
and the explicit form \cite{N4c=3,N4schoutens,N4KUL} of the currents
making up the $c=3$
algebra has been used. The average can be represented as a functional
integral with measure\col
\be
[d\hat{Q}][d\hat{P}]\exp \bigg( -{1 \over {2\pi}}(\hat{P}{\bar{\del} \over
\del}\hat{P}+\hat{Q}^a\bar{\del}\hat{Q}_a)\bigg) \quad. \label{eq:LNLgewicht}
\ee
The (nonlocal) form of the free action for $\hat{P}$ follows from its
two-point function\col\ it is the usual (local) free scalar field action if
one writes $\hat{P}=\del X$. The connection between the linear theory,
the nonlinear theory, and the $c=3$ realisation is then\col
\bea
\lefteqn{
   \exp \bigg( -{\pi\over k} \e^{abcd} \psi_a
      \dd{ \eta_b}\dd{ A_{cd}}\bigg)
      Z[\psi,A,\eta,b]\ =\ \widetilde{Z}[\psi,A] } \times
      \label{eq:LNLZtoZNL4}\\
&&
\exp \bigg( {{\pi^2}\over {3k^2}}(4+\sqrt{2k})  \e^{abcd}
\psi_a \dd{ \eta_b}\dd{ \eta_c}\dd{ \eta_d}\bigg)
Z^{c=3}[h,\psi,A,\eta\sqrt{k/2},b\sqrt{k/2}]  .\nonumber
\eea
Contrary to the $N=3$ case, where the Polyakov partition function was obtained
very explicitly, this connection is not particularly useful, but the
representation (\ref{eq:LNLgewicht}) of $Z^{c=3}$ as a functional integral
can be used effectively. Indeed, when we take the Fourier transform of
\vgl{eq:LNLZtoZNL4}, \ie we integrate (\ref{eq:LNLZtoZNL4}) with\col
\be
\int [d\,h][d\,\psi][d\,A][d\,b][d\,\eta]\exp \bigg( {1 \over \pi }
    \int\Bigl( h\,t +\psi_a\,g^a + A_{ab}\, u^{ab}  + b\, p +
       \eta_a\,q^a\Bigr)\bigg) \, ,
\ee
we obtain using eqs.\ \re{eq:LNLZetdrie} and \re{eq:LNLgewicht}\footnote{
The effective action $W$ is defined by the Fourier transform of $Z$, and
similarly for $\tW$.}\col
\bea
\lefteqn{
\exp \bigg( -W[t,\,g^a-{1\over k}\e _{abcd}q^bu^{cd},\,u,\,p,\,q] \bigg) =
\exp \bigg( -{1\over{\pi k}}(p{\bar{\del}\over \del}p+
                     q^a\bar{\del}q_a) \bigg)}\nonu
&&\exp \bigg(-\tW[t+{1\over k}\left(p^2+\del q\,q \right),\,
g^a+{2\over k}p\,q^a-{4\over {3k^2}} \e _{abcd}q^bq^cq^d,\!
u^{ab}-{2\over k}q^aq^b] \bigg) \nonu
&&
\eea
giving the concise relation\col
\bea
\tW[t,\,g^a,\,u^{ab}]
&+&{1\over{\pi k}}(p{\bar{\del}\over \del}p+q^a\bar{\del}q_a)\nonu
&=&W[t-{1\over k}\left(p^2+\del q^a\,q_a \right),\nonu
&&\qquad  g^a-{2\over k}pq^a-{1\over k}\e _{abcd}q^bu^{cd}
                 -{2\over {3k^2}}\e _{abcd}q^bq^cq^d,\nonu
&&\qquad  u^{ab}+{2\over k}q^aq^b,\,p,\,q^a]\,. \nonu
\eea
Putting the free $p$ and $q^a$-currents equal to zero, we obtain
\re{eq:FFWrelation}.

\section{Discussion}
%Let us consider the case of a conformal OPA where the systems that we
%decouple have the expected conformal dimension. For the \emt, the
%projection has to amount to substracting the usual \emt\ of the free fields.
%This can be checked using the formulas of section \ref{sct:FFalgorithms}.
%The new \emt\ is again a good Virasoro tensor with a central charge $c$
%shifted by minus the central charge of the decoupled system.  The other
%fields of the theory may become nonprimary after projection, but it should
%always be possible to find a new primary basis. We will not go further into
%this.

We have given explicit algorithms for the factoring out of free fields,
including a simple criterion for the factorisation of $U(1)$--scalars.
We have worked purely at the quantummechanical level,
but the algorithms do not need any modification for the
classical case if we use the definition \vgl{eq:PBregdef} for the
negative modes. In fact,
recently \cite{FORT,ragoucy} a number of classical {\WA}s were constructed by
hamiltonian reduction, containing bosons of dimensions $1$ and $\frac{1}{2}$
that could be decoupled. \puzzle{see p. 36, sect 3.2.2
Alex, p39 3.2.3}

Recently \cite{factorKA}, it was shown that
factoring out all dimension $1$ fields of a classical \WA\ gives rise to
algebras which are finitely generated by rational polynomials, \ie fractions of
generators are allowed. When regarding the fractions as new generators, a
nonlinear \WA\ with an infinite number of generators results.
In the quantum case only a finite number of generators are needed, due to
the appearance of null fields.

The examples in section \ref{sct:FFexamples} have shown that factorising a
linear algebra gives in general rise to nonlinear algebras. Conversely,
one could attempt to linearise a \WA\ by adding free fields and performing
a basis transformation. This was achieved for $W_3$ and the bosonic $N=2$
superconformal algebra in ref.~\cite{Sergey:linearWA}. Some important
features of the construction are that the free fields are not
quasiprimary with respect to the original \emt, and that the linear
algebra contains a ``null'' field $G$ in the sense that it has no OPE where
$\unity$ occurs. In fact, for $W_3$ the field $G$ is equal to $W-W_J$,
where $W_J$ is the expression of the one scalar realisation of $W_3$.
Clearly, this linear algebra is only of interest if one does
not put $G$ to zero.

The results of this chapter can be used to see if the effective action
of a nonlinear theory is related to its classical limit by introducing some
renormalisations, see section \ref{sct:RNsemiclass}.
If we take for granted that this is true for the linear theory,
\vgl{eq:FFWrelation} immediately transfers this property to the nonlinear
theory. Moreover, since the 'classical' parts are equal also (as implied by
the $c \rightarrow \infty$ limit of the Ward identities) the renormalisation
factors for both theories are the same (for couplings as well as for fields)
if one takes into account the shifts in the values for the central extensions
$c$, $ k_+$ and $k_-$. This fact can be confirmed by looking at explicit
calculations of these renormalisation factors, see chapter
\ref{chp:renormalisations}.




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%1cm is ongeveer 2.5 em

\def\db{\bar{\partial}}
\def\dz{\partial}
\def\e{{\varepsilon}}
\def\zb{{\bar{z}}}
\def\wb{{\bar{w}}}
\def\hb{{\bar{h}}}
\def\Tb{{\bar{T}}}
\def\ie{{\sl i.e.\ }}
\def\eg{{\sl e.g.\ }}
\def\rhs{{\sl rhs}}
\def\lhs{{\sl lhs}}
\def\col{{\nolinebreak :}}
\def\Mathematica{{\sl Mathematica}}
\def\OPEdefs{{\sl OPEdefs}}
\def\emt{en\-er\-gy--mo\-men\-tum tensor}
\def\emo{en\-er\-gy--mo\-men\-tum operator}
\def\WA{$\Ww$--al\-ge\-bra}
\def\WS{$\Ww$--string}
\def\KA{Ka\v{c}--Moody al\-ge\-bra}
\def\WZW{WZNW--mo\-del}
\def\capitel#1{\par\noindent
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\def\qed{\rule{1ex}{1ex}}

%OPEdefs stuff
\newcounter{mathline}
\newcounter{mathref}
\newcounter{mathreftwo}
\def\un{\symbol{95}}
\def\ha{\symbol{94}}
\def\lb{\char'173}
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    \makebox[\tab][l]{\sl Out[\themathline] =}{\tt #1}}
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    \makebox[\tab][l]{}{\tt #1}}
\newcommand{\contmind}[1]{\hspace*{9pt}
    \makebox[\tab][l]{}{\hspace*{.8cm} \tt #1}}

% Calligraphic letters

\def\ca{{\cal A}}
\def\cb{{\cal B}}
\def\cc{{\cal C}}
\def\cd{{\cal D}}
\def\ce{{\cal E}}
\def\cf{{\cal F}}
\def\cg{{\cal G}}
\def\ch{{\cal H}}
\def\ci{{\cal I}}
\def\cj{{\cal J}}
\def\ck{{\cal K}}
\def\cl{{\cal L}}
\def\cm{{\cal M}}
\def\cn{{\cal N}}
\def\co{{\cal O}}
\def\cp{{\cal P}}
\def\cq{{\cal Q}}
\def\car{{\cal R}}
\def\cs{{\cal S}}
\def\ct{{\cal T}}
\def\cu{{\cal U}}
\def\cv{{\cal V}}
\def\cw{{\cal W}}
\def\cx{{\cal X}}
\def\cy{{\cal Y}}
\def\cz{{\cal Z}}

%extra characters
\let\sss=\l
\mathchardef\Ww="3257 %script W

%\def\vu{\varepsilon}
%\def\a{\alpha}
%\def\b{\beta}
%\def\c{\chi}
\def\d{\delta}
%\def\e{\epsilon}                % Also, \varepsilon
%\def\f{\phi}                    %       \varphi
\def\g{\gamma}
\def\h{\eta}
%\def\i{\iota}
\def\j{\psi}
\def\k{\kappa}                  % Also, \varkappa (see below)
\def\l{\lambda}
\def\m{\mu}
%\def\n{\nu}
\def\o{\omega}
\def\p{\pi}                     % Also, \varpi
\def\q{\theta}                  %       \vartheta
%\def\r{\rho}                    %       \varrho
%\def\s{\sigma}                  %       \varsigma
\def\t{\tau}
%\def\u{\upsilon}
%\def\x{\xi}
%\def\z{\zeta}
%\def\D{\Delta}
%\def\F{\Phi}
\def\G{\Gamma}
%\def\J{\Psi}
%\def\L{\Lambda}
%\def\O{\Omega}
\def\P{\Pi}
%\def\Q{\Theta}
%\def\S{\Sigma}
%\def\U{\Upsilon}
%\def\X{\Xi}


\newcommand{\wha}{\widehat{\cal A}}
\newcommand{\wta}{\widetilde{\cal A}}
%\newcommand{\la}{\lambda}
%\newcommand{\Ga}{\Gamma}
%\newcommand{\de}{\delta}
\newcommand{\nab}{\overline{\nabla}}
\newcommand{\osp}{OSp(N|2)}
\newcommand{\asp}{osp(N|2)}
%\newcommand{\bz}{\bar{z}}
\newcommand{\slt}{sl(2)}
\def\soN{so(N)}
%\newcommand{\slt}{sl(2,{\bf R})}
\newcommand{\sln}{sl(n,{\bf R})}
\newcommand{\unn}{\underline{n}}

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\def\eeastar{\end{eqnarray*}}
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%
%  definition of \underhook{...}
%
\def\lefthook{{\vrule height5pt width0.4pt depth0pt}}
\def\righthook{{\vrule height5pt width0.4pt depth0pt}}
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%look for FULL?? if you change the [] convention
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\newcommand{\MO}[2]{{#1}_{#2}}
%OPE
\def\ope#1{<\!\!<#1>\!\!>}
\def\opetwo#1#2{<\!\!<#1\ |\ #2>\!\!>}
\def\opethree#1#2#3{<\!\!<#1\ |\ #2\ |\ #3>\!\!>}
\def\opefour#1#2#3#4{<\!\!<#1\ |\ #2\ |\ #3\ |\ #4>\!\!>}
\def\opefive#1#2#3#4#5{<\!\!<#1\ |\ #2\ |\ #3\ |\ #4\ |\ #5>\!\!>}
%binomial
\newcommand{\bin}[2]{\left(\!\!\!\begin{array}{c}#1\\#2\end{array}\!\!\!\right)}
\def\Pochhammer#1#2{\left(#1\right)_{#2}}
%notation for (quasi)primary- coefficients
\def\QPc#1#2#3#4{\alpha({#1},{#2},{#3},{#4})}
\def\QPctwo#1#2{\alpha({#1},{#2})}
\def\Pc#1#2#3#4{\beta({#1},{#2},{#3},{#4})}
\def\Pctwo#1#2{\beta({#1},{#2})}
\def\Pqpc#1#2#3#4{\tilde{\beta}({#1},{#2},{#3},{#4})}
\def\Pqpctwo#1#2{\tilde{\beta}({#1},{#2})}
\def\QPs{{\eul a}}

\newcommand{\ef}{{\rm eff}}
\newcommand{\ind}{{\rm ind}}
%\newcommand{\var}{\varphi}
\newcommand{\del}{\partial}
\newcommand{\delb}{\bar{\partial}}
\newcommand{\bdel}{\bar{\partial}}
\newcommand{\nablab}{\overline{\nabla}}
%\newcommand{\DZ}{\partial_z}
%\newcommand{\zw}{(z-w)}
\newcommand{\pp}{=\!\!\! |}
\def\ppmm{{\stackrel{\scriptstyle\!\! \pp}{=}}}
\newcommand{\str}{\mbox{\em str}}
\newcommand{\sdet}{\mbox{\em sdet}}
\def\KER#1{{\ck_{#1}}}
\def\PHW{\P_{hw}}
\def\PLW{\P_{lw}}

\newcommand{\sectionappendix}{
\renewcommand{\thesection}{\thechapter.\Alph{section}}%
\setcounter{section}{0}}
\newcommand{\sectionnormal}{
\renewcommand{\thesection}{\thechapter.\arabic{section}}
   \setcounter{section}{0}}

%\hyphenation{ener-gy--momen-tum}

%\def\puzzle#1{{\newline \sc ?? #1}}
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%\includeonly{factfree}
%\includeonly{wznw}
%\includeonly{renorm}
%\includeonly{wstring}
%\includeonly{appendix}
%\includeonly{biblio}
%\includeonly{samen}
%\includeonly{test}
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\include{title}
%\include{titlebij}
\pagenumbering{roman}
\pagestyle{krisheadings}
\setcounter{page}{1}
%\parskip10pt plus 1pt minus 1pt
\tableofcontents
%\parskip5pt plus 1pt minus 1pt
\ifdankwoord
\include{notation}
\fi
\def\theequation{\thechapter.\arabic{equation}}
\pagenumbering{arabic}
\pagestyle{krisheadings}
\setcounter{page}{0}
\include{intro}
\def\theequation{\thesection.\arabic{equation}}
%\def\thepage{\arabic{page}}
\include{cft}
\include{opedefs}
\include{confinv}
\include{factfree}
\include{wznw}
\include{renorm}
\include{wstring}
\def\theequation{\thechapter.\arabic{equation}}
\include{appendix}
%\include{test}
%\pagenumbering{roman}
\include{biblio}
\def\theequation{\arabic{equation}}
\ifdankwoord
\include{samen}
\fi
\end{document}

\chapter[OPEs in Mathematica]{Operator Product Expansions in
Mathematica \label{chp:OPEdefs}}
\mychapter
The previous chapter indicates that the method of Operator Product
Expansions (OPEs) gives us a very useful algebraic tool to study a
conformal field theory. However, the relevant formulas show that
calculations can become quite tedious and error-prone, especially when
nested normal ordered products are involved. This was the motivation to
implement an algorithm to handle OPEs and normal ordered products. By now,
the \OPEdefs\ package is widely used, showing that the need for
automated OPE calculations was (and is) present among conformal field
theorists.

This chapter discusses the \OPEdefs\ package.
In the first section, the desired capabilities of the package are
discussed and a brief history of the development of \OPEdefs\ is given.
Section \ref{eq:OPEdesign} is devoted to design considerations. The
implementation we used is explained in section \ref{sct:OPEimplementation}.
We first give the algorithm, and prove that it is finite. Then some
crucial points in the actual code of \OPEdefs\ are highlighted.
Section \ref{sct:OPEdefsGuide} is a user's guide to \OPEdefs, and
the next section presents an explicit example of a calculation.
Section \ref{sct:OPEfuture} gives an outlook on possible extensions.
We end this chapter with a review of other symbolic manipulation packages
used in conformal field theory.

Some of the material found in this chapter was published in
\cite{OPEdefs2.0} and \cite{OPEdefs3.0}.

\begin{center}
\tiny
All registered trademarks used in this chapter are acknowledged.
\end{center}

\section{Intention and history}
We want to construct a package which, given the OPEs of a set of
generators, is able to compute any OPE in the Operator Product Algebra
(OPA)\footnote{When we say, ``compute an OPE'', we mean computing all
operators of the singular part.}.
Moreover, as the definition of normal ordering is noncommutative
and nonassociative, the package should bring composite operators in a
standard form. This can be done by introducing an order for the
generators and reordering composites such that generators are ordered
from left to right, using the associativity conditions of the previous
chapter.

We will restrict ourselves to meromorphic conformal field theory.
As we use only the concept of an OPA (see def. \ref{def:OPA}),
we will be able to use operators with any (also negative)
conformal dimension.

An important step in the calculations with OPEs is to check that the
Jacobi identitities \re{eq:OPEJacRAB} discussed in the previous
chapter are satisfied. The package should be able to verify this. This
will also enable us to fix some free parameters in the OPEs of the
generators by requiring associativity.

A first version of my package was used to perform the calculations on
the Casimir algebra of $B_2$ \cite{wb2} (see
section \ref{sct:WB2}), where OPEs of composites of up to four fields are
involved. After some optimisations, \OPEdefs\ 2.0
\cite{OPEdefs2.0} was made available to the public. Several well-known
results were checked with this version, \eg the Sugawara construction
for $\widehat{SU(n)}$, the free field realisations and homological
constructions of $W_3$ based on a $\widehat{SU(3)}$-\KA\ \cite{su3}. The
power of having all calculations automated was already clear at this
point, as we found a small misprint in \cite{su3}\footnote{The second
term in the expression for $J_3^+$ (eq. 2.15) of \cite{su3}  should
read $-\beta_1 \gamma_1\gamma_3$.}, while this calculation was highly
nontrivial by hand.

Important improvements (version 3.0) were published in \cite{OPEdefs3.0}.
It was necessary to introduce a different syntax, to be able to make
extensive changes in the future. Since then, many other small changes
have been made to the package. The latest version, which will be
discussed in this chapter, is 3.1.

The package is freely available from the author\footnote{It should be
in the {\sl MathSource} archive soon. See\\
{\tt http://www.wri.com/mathsource/index.html}.}.

\section{Design considerations\label{eq:OPEdesign}}
In this section, we comment on how \OPEdefs\ was designed.
Several aspects are important.
\begin{itemize}
\item How will the program present itself to the user ? (interface)\\
Preferably, concepts and terminology that the user has to handle
should be familiar to any conformal field theorist.

\item How will the data be represented ?\\
The representation has to be chosen such that storage space and
computation time are minimised. Furthermore, it should be possible to
incorporate any related problems (for instance Poisson brackets) without
changing the data representation.\\
Whatever representation is chosen, it is important
to hide this choice for the user. Ideally, one should be able to change
the representation (and thus the internal working of the program)
without affecting the interface.

\item Make sure the program can be easily extended.\\
It is important to keep the main ideas of the algorithm reflected in
the structure of the program, separated from the actual details of the
computation. This implies a modular style of writing. Of course, a high
level programming language enhances readability and makes changes and
additions easier.

\item How can the results be manipulated?\\
It is of course not enough to be able to compute a certain OPE or normal
ordered product. Different results have to be added, coefficients
simplified or extracted, equations solved\ldots\ It would lead the
programmer of the package way to far if he/she had to implement all this
functionality.
\end{itemize}

These remarks point towards an implementation in an interactive
environment where it is possible to perform symbolic computations, and
where a high level language is available, \ie in a symbolic manipulation
program. Indeed, these environments are developed to perform the dull
-- or algorithmic -- calculations, once the necessary rules are
implemented. The user can then use the programming power of the
environment to solve his problem.
It would be perfectly possible to write the proposed program in an
``ordinary'' programming language like {\sl $C^{++}$} or {\sl lisp}. After
all, the symbolic manipulation programs are written in one of these
languages. However, the amount of code required would probably go beyond
the scope of a physics PhD.

Many of the design principles we mentioned, are reminiscent
of object oriented programming. As this is a fairly new concept, one
immediately is directed towards the more recent environments. Of
these, the best known and more powerful are {\sl Axiom} \cite{Axiom}, {\sl
Maple} \cite{Maple} and \Mathematica\ \cite{Wolfram}. All three are
suitable for implementation of the algorithm for working with OPEs. \\
From the viewpoint of the programmer, {\sl Axiom} seems to be the most
attractive choice. It is indeed the most recent environment, and its
design reflects all points made above. The major disadvantages of
this system are that it is not very popular yet, and the hardware
requirements are very high (a powerful Unix workstation is needed).\\
{\sl Maple} is being used more and more over the last few years. Its
design is concentrated on run-time efficiency and small memory
requirements. Presumably, a consequence of this is that the {\sl Maple}
programming language is more like $C$ or {\sl Pascal}, although with powerful
extensions for symbolic manipulations. However, it does not seem to
permit a high abstraction level and modularity compared to the other two
environments.\\
Finally, the \Mathematica\ environment is somewhere in between the two
others. It is probably not as efficient as {\sl Maple}, but it is a lot
easier to program in. It runs on a lot of machines, from PC's to
supercomputers and all versions are completely compatible. Many of the
ideas of object oriented programming can be used in \Mathematica\ and its
pattern matching capabilities are very useful for any mathematical
programming.

\section{Implementation\label{sct:OPEimplementation}}
In this section, we first discuss the algorithm which we used. In
principle, it simply consists of repeated application of the
rules given in subsection \ref{ssct:OPErules}. In the next subsection, we
give some parts of the actual code of \OPEdefs. Finally, we comment on the
performance of the package.

\subsection{Algorithm\label{ssct:OPEAlgorithm}}
We will discuss the algorithm in two steps. First, we show how the
singular part of the OPEs can be computed. Then we give the algorithm for
simplifying composites. In \OPEdefs, composites are simplified at each
step of the calculation.

\OPEdefs\ 3.1 can calculate OPEs and Poisson brackets. We concentrate on
OPEs here and briefly comment which changes are needed for the other case.

\subsubsection{Computing OPEs}
The package should compute OPEs of arbitrarily complicated composites
when a set of generators and their OPEs is given. Obviously, we will need
the rules constructed in subsection \ref{ssct:OPErules}. For easy
reference, we list the rules we need, aside from linearity.
First, there are the rules involving derivatives\col
\bea
[\dz A\,B]_{q}& =& -(q-1) [AB]_{q-1} \label{eq:RulePolederL}\\[2mm]
[A\,\dz B]_{q}& =& (q-1) [AB]_{q-1} + \dz [AB]_{q}\,.
    \label{eq:RulePolederR}
\eea
Next, we need the OPE of $B$ with $A$, given the OPE of
$A$ with $B$\col \be
[BA]_q = (-1)^{|A||B|}
\sum_{l\geq q}{\frac{(-1)^l}{(l-q)! } \partial^{(l-q)}[AB]_l}
\label{eq:RuleOPEcomm}\,.
\ee
OPEs with composites can be calculated using \vgl{eq:OPEJacRAB}\col
\bea
[A\ [BC]_0]_q &=& (-1)^{|A||B|} [B\ [AC]_q]_0 + [[AB]_q\ C]_0\nonu
&& + \sum_{l=1}^{q-1} \bin{q-1}{l} [[AB]_{q-l}C]_l
\label{eq:RuleOPEcompRAB}
\eea
where $q \geq 1$. 

These rules are the only ones needed to compute every OPE in the OPA.
Indeed, when computing the OPE of $A$ with $B$, we apply the following
procedure\col
\begin{itemize}
\item if $A$ and $B$ are generators whose OPE we know, return it as
   the result.
\item apply linearity if necessary.
\item if $A$ is an operator with derivatives, use \vgl{eq:RulePolederL}.
\item if $B$ is an operator with derivatives, use \vgl{eq:RulePolederR}.
\item if $B$ is a composite, use \vgl{eq:RuleOPEcompRAB}.
\item if $A$ is a composite, use \vgl{eq:RuleOPEcomm}.
\item if the OPE $B(z)\ A(w)$ is known, compute the OPE $A(z)\ B(w)$ using
  \vgl{eq:RuleOPEcomm}.
\end{itemize}
This list should be used recursively until none of the rules applies,
which means that the OPE has been calculated. The order in which we check
the rules is in this case not important, but we will check them in a
``top-down'' order. Note that to compute an OPE of a composite with a
generator, first \vgl{eq:RuleOPEcomm} is used, and
\vgl{eq:RuleOPEcompRAB} in the next step. The question now is if this is
a finite procedure.

First of all, all sums in the previous equations have a
finite number of terms. This follows from the assumption that all OPEs
between generators in the OPA have a finite number of poles. The only
possible problem is that \vgl{eq:RuleOPEcompRAB} would require an
infinite number of steps. Indeed, when computing the OPE $\SING{A\
{[}B}C]_0$, one needs all OPEs $[A\SING{B{]_n}\ C}$ for $n>0$. It is now
possible that $[AB]_n$ contains a composite again, leading to a new
application of \vgl{eq:RuleOPEcompRAB}. For graded OPAs (see def.
\ref{def:dimension}), we can prove the following theorem.
\begin{theorem}\label{trm:OPEAlgorithm}
If the OPA is graded and there is a lower bound
$D$ on the dimension of all the operators in the OPA, then
the application of eqs.\
(\ref{eq:RulePolederL}-\ref{eq:RuleOPEcompRAB}) requires only a finite
number of steps.
\end{theorem}
\begin{proof}
The sum of the dimensions of the fields in the OPE
$\FULL{A_1\ [B_1C_1]_0}$ is $\Delta=a_1+b_1+c_1$. The application of
\vgl{eq:RuleOPEcompRAB} leads to the computation of the
OPEs $\FULL{[AB]_n\ C}$ for $n>0$. The sum of the dimensions is in
this case $a_1+b_1+c_1-n$. As this is strictly smaller than $\Delta$,
we see that after (the integer part of) $\Delta-2D$ steps, we would need
the OPE of a field with dimension lower than $D$, which is not present
in the OPA by assumption.
\end{proof}
Note that the assumption of the existence of a lower bound on the
dimension is not very restrictive. First of all, all conformal
OPAs (see def. \ref{def:COPA}) with generators of positive dimension
trivially satisfy the criterion. Moreover, a finite number of fermionic
generators can have negative dimension. Indeed, to form composites of negative
dimension with these fermions, one needs extra derivatives -- 
\eg $[\dz A\ A]_0$ -- imposed by \vgl{eq:Rulecompfermion} below.

An important remark here is that the dimension used in the proof is not
necessarily the conformal dimension of the operators. For instance, for a
bosonic $bc$-system which commutes with all the other generators of the
OPA, we can assign a dimension of $1/2$ to both $b$ and $c$, such that
the theorem can be used.

The algorithm used to compute Poisson brackets is essentially the same.
Some small changes in the rule \vgl{eq:RuleOPEcompRAB}, as discussed in
subsection \ref{ssct:Poisson}, are needed. 

\subsubsection{Simplifying composites}
We now discuss how we can reduce normal ordered products
to a standard form. We define an order on the generators and their
derivatives, \eg lexicographic ordering. Given a composite, we apply the
rules of subsection \ref{ssct:OPErules} until all composites are normal
ordered from right to left and the operators are ordered, \ie
$[A[B[C\ldots]_0]_0]_0$, and $A\leq B\leq C$. The relevant formulas
are\col
\bea
&&\dz [AB]_0= [A\ \dz B]_0 + [\dz A\ B]_0 \label{eq:RuleNOder}\\[2mm]
&&[AB]_{-q} = {1\over q!} [(\dz^q \!\!A)\ B]_0\,,\qquad\qquad q\geq 1
\label{eq:RulePoleregdef} \\[2mm]
&&[BA]_0 = (-1)^{|A||B|} [AB]_0 + (-1)^{|A||B|}
\sum_{l\geq 1}{\frac{(-1)^l}{l! } \partial^{l}[AB]_l}
\label{eq:RuleNOcomm}\\[2mm]
&&[AA]_0 = -\sum_{l>0}{(-1)^{l}\over 2l!} \partial^l [AA]_{l}
\ ,\ \ \ \mbox{for $A$ fermionic .}
\label{eq:Rulecompfermion}\\[2mm]
&&[A\ [BC]_0]_0 = (-1)^{|A||B|} [B\ [AC]_0]_0 +\\[2mm]
&&\qquad\qquad [([AB]_0 -
(-1)^{|A||B|}[BA]_0)\ C]_0 \label{eq:RuleNOcompR}\\[2mm]
&&[A\ [AC]_0]_0 = [[AA]_0 \ C]_0 
\ ,\ \ \ \mbox{for $A$ fermionic .}\label{eq:RuleNOfermion}
\eea
The rule \re{eq:RuleNOcompR} follows from \vgl{eq:OPEJacRAB}.

We have to wonder if we again require only a finite number of steps to
reach the end-result. We can not prove this along the same lines as
theorem \ref{trm:OPEAlgorithm}. Indeed, in \vgl{eq:RuleNOcompR} all terms
necessarily have the same dimension. We give an
example of infinite recursion in the intermezzo.
\begin{intermezzo}
Consider an OPA with bosonic operators $A,B$ such that\col
\be
[AB]_1=a B\,.\label{eq:OPEnullOPE}
\ee
We do not specify the OPE of $B$ with itself. We want to reorder
$[[AB]_0B]_0$ into standard order, \ie $[A[BB]_0]_0$ + corrections. In
the computation which follows, an ellipsis denotes terms containing more
than one derivative or containing $[BB]_n$ with $n>0$. (The equation
numbers at the right correspond to the formula used in a certain step.)
\[
\begin{array}{llr}
[[AB]_0B]_0\!\! &=\ [B[AB]_0]_0 - \del [B[AB]_0]_1 + \ldots&
      \vgl{eq:RuleNOcomm}\\
&=[A[BB]_0]_0 + [\del[BA]_1\ B]_0-& \vgl{eq:RuleNOcompR} \\
&\ \ \ \del\left([A[BB]_1]_0+[[BA]_1B]_0\right) + \ldots&
      \vgl{eq:RuleOPEcompRAB}\\
&=[A[BB]_0]_0 -
 a [\del B\ B]_0 + a \del[BB]_0+\ldots&
      eqs.\ \re{eq:RuleNOder},\re{eq:RuleNOcomm}\\
&=[A[BB]_0]_0 + a [\del B\, B]_0 + \ldots&
\end{array}
\]
Now, suppose that there is a null operator\col
\be
\del B - {1\over a}[AB]_0 =0\,.\label{eq:OPEnull}
\ee
in the OPA and we use this relation to eliminate $\del B$. We
find\col
\[ [[AB]_0B ]_0 = [A[BB]_0]_0 + [[AB]_0B]_0 +\ldots \,,\]
which means that we cannot determine $[[AB]_0B ]_0$ in this way.
Note that there is no problem if we use \vgl{eq:OPEnull} to eliminate
$[AB]_0$.

This example occurs for a scalar field $X$ and the vertex operators
defined in subsection \ref{ssct:FreeScalar}. Take $A=\del X$ and $B=:\exp
(a X):$. If we normalised $X$ such that $[\del X\, \del X]_2=1$,
\vgl{eq:OPEnullOPE} is satisfied. On the other hand, $\del (:\exp (a X):) =
a [\del X\ :\exp (a X):]_0$. Clearly, this corresponds to
\vgl{eq:OPEnull} if $a=1$.
\end{intermezzo}
The example in the intermezzo shows that infinite recursion can occur when
null operators are present. If $N$ is a null operator, we call $N=0$ a
null-relation. As shown in the intermezzo, the algorithm does not work if
we use a null-relation to eliminate operators with ``less'' derivatives
in favour of those with ``more'' derivatives.

We make this more precise. For an operator $A$, we define $d(A)$ as the
number of derivatives occuring in $A$. When we do not use any
null-relations, we have\col
\bea
d(A)&=&0\hspace*{4em}{\rm if}\ A\ \mbox{is a generator}\nonumber\\
d(\dz A)&=&d(A)+1\nonumber\\
d([A B]_0)&=& d(A)+d(B)\nonumber\\
d(A+B) &=& \min(d(A),d(B))\,
.\label{eq:OPEdn}
\eea
We say that we ``use a null-relation in increasing derivative-number'' if we
use it to eliminate (one of) the operator(s) with the smallest
derivative-number. In this case, two of the relations \vgl{eq:OPEdn} are
converted to inequalities\col\ $d(\dz^n A)\geq n$ and $d([A B]_0)\geq
d(A)+d(B)$.

We are now in a position to prove the following theorem.
\begin{theorem}\label{trm:NOAlgorithm}
In an OPA satisfying the conditions of theorem \ref{trm:OPEAlgorithm},
and where any null-relations are used in increasing derivative-number,
the application of eqs.~(\ref{eq:RuleNOder}-\ref{eq:RuleNOfermion})
(together with eqs.\ (\ref{eq:RulePolederL}-\ref{eq:RuleOPEcompRAB}))
to order the operators in a normal ordered product requires only a finite
number of steps.
\end{theorem}
\begin{proof}
We define\col
\[
n\equiv d\left([A [B C]_0]_0\right)\,.
\]
We wish to show that the derivative number of the correction terms in
\vgl{eq:RuleNOcompR} is always bigger than $n$\col
\[ d\left([A[BC]_0]_0 - (-1)^{|A||B|}[B[AC]_0]_0\right) \ >\ n\,.
\]
When $n=0$ this is immediately clear.\\
For $n=1$ we have to consider the different orderings. Let us look first
at the case where $A=\del X$\col
\bea
[\dz X [B C]_0]_0\!\! &=&\!\!
[B [\dz X\ C]_0]_0 - \sum_{l\geq 1} {(-1)^l\over l!}
   [\dz^l [\dz X\ B]_l\ C]_0\nonu
&=&\!\![B [\dz X\ C]_0]_0 + \sum_{l\geq 2}
   {(-1)^l (l-1)\over l!} [\dz^l[X B]_{l-1} \ C]_0\,
   ,\label{eq:OPENOproof}
\eea
where we used \vgl{eq:RulePolederL} in the last step. We see that the
derivative-number for the second term in the \rhs\ is at least $2$. The
case where $B=\del X$ follows from \vgl{eq:OPENOproof} by moving the
correction terms to the \lhs. Finally, when $C=\del X$, the correction
terms will be of the form $[\dz^l[AB]_l\dz X]_0$, which again have
derivative number greater than $2$ because $l\geq 1$.

It is now easy to see that the same reasoning can be used for
general $n$. For this we will need that\col
\[ d(\dz^l [\dz^m\! A\ \dz^n\! B]_l)\ \geq\ m+n+1\, , \]
(or $[\dz^m\! A\ \dz^n\! B]_l=0$) which follows from eqs.\
\re{eq:RulePolederL} and \re{eq:RulePolederR}.

To conclude the proof, we note that with each application of
\vgl{eq:RuleNOcompR} the derivative-number of the correction terms
increases. On the other hand, the dimension of the correction terms
remains constant. However, for any operator $A$, one has that\col
\[ d(A) \leq {\rm dim}(A) - D \,,\]
where $D$ is the smallest dimension occuring in the OPA.
This shows that after a finite number of applications of
\vgl{eq:RuleNOcompR}, no correction terms can appear.
\end{proof}

Theorem \ref{trm:NOAlgorithm} guarantees that the algorithm to
reorder composites ends in a finite number of steps.
However, we have not yet proven that the end-result of the algorithm gives
a ``standard form'', \ie is the result unique if two expressions are the
same up to null-relations?
\begin{intermezzo}
Consider the following counterexample. Suppose there is a null-relation\col
\[
[AB]_0-C=0\,,
\]
where $A,B,C$ are bosonic operators. If we use this relation to
eliminate the composite $[AB]_0$, we find immediately
\[
[[AB]_0D]_0 = [CD]_0\,,.
\]
On the other hand, two applications of \vgl{eq:RuleNOcomm} and one of
\vgl{eq:RuleNOcompR} show that
\[
[[AB]_0D]_0 = [A[BD]_0]_0 + \ldots
\]
where the ellipsis denotes terms with derivatives. Together, these
relations show that $[A[BD]_0]_0 = [CD]_0 + \ldots$ while both \lhs\ and
\rhs\ are not changed by the algorithm.
\end{intermezzo}
The intermezzo shows that null-relations should be used to eliminate the
``simpler'' fields in terms of the composites.
We need the notion of ``composite-number'' $c(A)$. When we do not use any
null-relations, we have\col
\bea
c(A)&=&1\hspace*{4em}{\rm if}\ A\ \mbox{is a generator}\nonumber\\
c(\dz A)&=&c(A)\nonumber\\
c([A B]_0)&=& c(A)+c(B)\nonumber\\
c(\sum_iA_i) &=&
       \min_{\forall i\ {\rm such\ that}\ d(A_i)=d(\sum_jA_j)}c(A_i)\,.
\label{eq:OPEcn}
\eea
With these definitions, the composite-number is equal for both sides of
all eqs.~(\ref{eq:RuleNOder}-\ref{eq:RuleNOcompR}).
We say that we ``use a null-relation in increasing derivative- and
composite-number'' if we
use it to eliminate (one of) the operator(s) with the smallest
composite-number of those with the smallest
derivative-number. Note that we have $c(\dz A)\geq c(A)$ in this case.
We can now assert\col
\begin{theorem}\label{trm:NOAlgorithmUnique}
In an OPA satisfying the conditions of theorem \ref{trm:OPEAlgorithm},
and where any null-relations are used in increasing derivative-
and composite-num\-ber,
the application of eqs.~(\ref{eq:RuleNOder}-\ref{eq:RuleNOfermion})
(together with eqs.\ (\ref{eq:RulePolederL}-\ref{eq:RuleOPEcompRAB}))
to order the operators in a normal ordered product defines a ``standard
form'' on the elements of the algebra.
\end{theorem}

For the case of Poisson brackets, the rules \re{eq:RuleNOcomm} and
\re{eq:RuleNOcompR} drastically simplify, only the first terms remain.
Also, \vgl{eq:Rulecompfermion}
is changed to $[AA]_0=0$ when $A$ is fermionic.

\subsubsection{Improvements}
The rules given in this section up to now are sufficient to
compute any OPE, and to reorder any composite into a standard form.
However, some shortcuts exist. $[A\ [BC]_0]_q$ can be computed using
\vgl{eq:RuleOPEcompRAB}, but an alternative follows
from \vgl{eq:OPEJacRBA} using \vgl{eq:RulePoleregdef}\col
\bea
[A\ [BC]_0]_q &=&
 (-1)^{|A||B|}\Big( [B\ [AC]_q]_0 + \sum_{l\geq 0}{(-1)^{l+q}\over l!}
               [\dz^l[BA]_{l+q}\ C]_0\nonumber\\
&& \qquad\qquad +\sum_{l=1}^{q-1}(-1)^l [[BA]_l\ C]_{q-l}\Big)\,.
    \label{eq:RuleOPEcompRBA}
\eea
This rule is more convenient when we know the OPE $B(z)A(w)$ while
$A(z)B(w)$ has to be computed using \vgl{eq:RuleOPEcomm}.

Similarly, to compute an OPE where the first operator is a composite,
the algorithm as presented above uses eqs.\ \re{eq:RuleOPEcomm}
and \re{eq:RuleOPEcompRAB}.
Clearly, \vgl{eq:OPEJacL} and \vgl{eq:RulePoleregdef} implement
this in one step\col
 \bea [[AB]_0\ C]_q &=&
    \sum_{l\geq 0}{1 \over l!}  [\dz^l A\ [BC]_{l+q}]_0
   + (-1)^{|A||B|}\sum_{l\geq 0}{1 \over l!} [\dz^l B\ [AC]_{l+q}]_0
\nonumber\\
   & & + (-1)^{|A||B|} \sum_{l=1}^{q-1} [B\ [AC]_{q-l}]_,,
\label{eq:RuleOPEcompL}
\eea
where $q \geq 1$ and\col
\bea
[[AB]_0 C]_0 &=&
    [A[BC]_0]_0 +\nonumber\\
&&    \sum_{l>0} {1 \over l!} [\dz^l A\ [BC]_{l}]_0
    + (-1)^{|A||B|} \sum_{l>0}{1\over l!} [\dz^l B\ [AC]_{l}]_0 \,.
 \label{eq:RuleNOcompL}
\eea
We will see in subsection \ref{ssct:OPEdefsInternals} where these rules are
applied in \OPEdefs\ 3.1.

\subsection{The internals of {\sl OPEdefs}\label{ssct:OPEdefsInternals}}
\setcounter{mathline}{0}
In this subsection, some crucial points in the implementation of the program
are discussed. The rest of this chapter does not depend on the material
presented here.
Familiarity with the  \Mathematica\ programming
language and packages is assumed. More information concerning these
topics can be found in appendix \ref{app:Mathematica} and \cite{Wolfram}.
For clarity, some of the rules in this subsection are slightly
simplified versions of those appearing in the actual code of \OPEdefs.
Input for \Mathematica\ is written in {\tt typeset} font.

\subsection{Operator handling}
Before any calculations can be done, we have to distinguish between
operators and scalars. The user of the package has to declare the
bosonic and fermionic operators he/she want to use with the
functions {\tt Bosonic} and {\tt Fermionic} \eg
\begin{verbatim}
Bosonic[A,B]
\end{verbatim}
This declaration simply sets the appropriate values for some internal
functions\col\ a function which distinguishes between operators and
scalars ({\tt OperatorQ}), and another for bosons and fermions ({\tt
BosonQ}).
\bv
BosonicHelp[A_] :=   (BosonQ[A]=True; OperatorQ[A]=True;
                      OPEposition[A] = OPEpositionCounter++)
Bosonic[A__] := Scan[BosonicHelp,{A}]
OPEpositionCounter = 0
\end{verbatim}
and similar rules for {\tt Fermionic}. The function {\tt OPEposition}
is used to record the order between the operators, which is determined
by the order in which they were declared and the standard \Mathematica\
order for operators declared with the same pattern\col
\bv
OPEOrder[a_, b_] :=
    Block[{res = OPEposition[b] - OPEposition[a]},
        If[res == 0, Order[a, b], res]
    ]
\end{verbatim}
We now list all rules that are necessary for testing if an expression is an
operator or not.\\[2mm]
\begin{minipage}[b]{9cm}
\bv
OperatorQ[A_+B_] := OperatorQ[A]
OperatorQ[A_Times] :=
     Apply[Or, Map[OperatorQ, Apply[List,A]]
OperatorQ[Derivative[_][A_]] := OperatorQ[A]
OperatorQ[_NO] = True
OperatorQ[0] = True
OperatorQ[_] = False
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:OperatorQ}
\ee
\end{minipage}

The second rule is the most complicated one. It handles the testing
of a product by testing if any of the factors is an operator. The fourth
rule declares any composites (which we will give the head {\tt NO}) to be
operators. The last rule says that all expressions which were not handled
by the previous rules (and the rules set by any declarations) are
scalars.\\
Note that this simple way of defining {\tt OperatorQ} is only
possible because \Mathematica\ reorders the rules such that more general
rules are checked last. This means that when a user declares {\tt A} to
be bosonic, a {\tt OperatorQ[A]=True} will be added on top of the list of
rules given in \re{v:OperatorQ}.

The only rules we need for the function {\tt BosonQ} are given below,
together with the definition of {\tt SwapSign} which gives the sign when
interchanging two operators, \ie $-1$ for two fermions and $+1$
otherwise\footnote{{\tt Literal} prevents \Mathematica\ to simplify the
left hand side of a definition.}\col \bv
BosonQ[Derivative[_][A_]] := BosonQ[A]
Literal[BosonQ[NO[A_,B_]]]:= Not[Xor[BosonQ[A],BosonQ[B]]]
SwapSign[A_,B_] := If[Or[BosonQ[A], BosonQ[B]],1,-1]
\end{verbatim}

The unit operator $\unity$ is declared in \OPEdefs, and is written as
\verb|One|. Introducing an explicit unit operator considerably simplifies
many of the functions in \OPEdefs.

\subsubsection{Data representation}
The representation we used in \OPEdefs\ 2.0 was a Laurent series
representation. This allowed us to use the built-in rules of
\Mathematica\ to add OPEs, take derivatives with respect to one of the
arguments, $\ldots$. However, storage space turned out to be a problem
for complicated algebras. Therefore, we now use a representation for the
singular part of an OPE which is a list of the operators at the different
poles. The highest order pole occurs first in the list, the first order
pole is the last. As an example, the OPE of a Virasoro operator
with itself, \vgl{eq:VirasoroOPE}, is represented as\col
\bv
OPEData[{c/2 One, 0, 2T, T'}]
\end{verbatim}
This representation has the additional advantage that it is more
general than a Laurent series representation. Poisson brackets form an
obvious example, see subsection \ref{ssct:Poisson}.

To hide the details of the representation to the user, the
user-interface for constructing an OPE is the function {\tt MakeOPE}.
This enables us to allow a different syntax for the interface and the
representation, see section \ref{sct:OPEdefsGuide}.

With the representation we use, it is easy to determine the order of the
highest pole\col \bv
MaxPole[OPEData[A_List]] := Length[A]
\end{verbatim}
For the Virasoro OPE this obviously gives $4$.

The actual definitions for the {\tt OPEData} structure enable
us to multiply an OPE with a scalar, and add two OPEs. Also higher
order poles that are zero are removed. The addition turns out to
be the most complicated to achieve. One is tempted to use the {\tt
Listable} attribute of {\tt Plus}, \ie  adding two list of the same
length gives a list with the sum of the corresponding elements. To be
able to use this feature, we should make the lists of poles
the same length by extending the shorter list with extra zeroes
(corresponding to zero operators at those poles). As addition of OPEs
will occur frequently, the corresponding rule should be as
efficient as possible\col \bv
OPEData /: n_ * OPEData[A_List] := OPEData[n*A]
OPEData /: A1_OPEData + A2__OPEData :=
        Block[{maxP = Max[Map[MaxPole, {A1,A2}]]},
            OPEData[ Plus @@
                Map[Join[ Table[0,{maxP-Length[#]}], # ]&,
                    Map[First, {A1,A2}]
                ]
            ]
        ]
OPEData[{0, A___}] := OPEData[{A}]
\end{verbatim}
\begin{intermezzo}
In this intermezzo we explain the rule for the addition of {\tt
OPEData} structures. The \lhs\ of the definition makes sure that {\tt
\{A1,A2\}} will be the list of all {\tt OPEData}s which are
added\footnote{A similar effect would be obtained with a \lhs\
{\tt Literal[Plus[A\un\un OPEData]]}. However, this gives
an infinite recursion when one adds erroneously a constant to
an OPEData. Indeed, \verb|Plus[1, OPEData[{One}]| matches
{\tt Literal[Plus[A\un\un OPEData]]}, and \Mathematica\
would keep applying the rule, changing nothing at all).}. When adding a
number of {\tt OPEData} structures, {\tt maxP} is set to the order of the
highest pole occuring in the sum. Then {\tt First} extracts the lists of
poles, and we add the correct number of zeroes to the left. Finally, the
sum of all these lists is made and {\tt OPEData} wrapped around the
result. As an example, consider the sum of {\tt OPEData[\{T,0\}]} and
{\tt OPEData[\{T'\}]}, we get\col \bv
{A1,A2}          -> {OPEData[{T,0}], OPEData[{T'}]}
maxP             -> 2
First ...        -> {{T,0}, {T'}}
Join ...         -> {{T,0}, {0,T'}}
Plus @@ ...      -> {T,T'}
\end{verbatim}
\end{intermezzo}
\subsubsection{OPE rules}
We use the head {\tt OPE} to represent an OPE. The bilinearity of the
{\tt OPE} function is defined by\col\\[2mm]
\begin{minipage}[b]{9cm}
\bv
Literal[OPE[A___,s_ B_,C___]] :=
         s OPE[A,B,C] /; OperatorQ[B]

Literal[OPE[a___,b_Plus,c___]]:=
         Distribute[
                Lineartmp[a,b,c],
                Plus,Lineartmp,
                Plus,OPE
        ]
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:OPElinear}
\ee
\end{minipage}


\begin{intermezzo}
The rule for an OPE of a sum uses the
\verb&Distribute[f[__],g,f,gn,fn]& built-in function, which implements
distributivity of {\tt f} with respect to {\tt g}, replacing {\tt g}
with {\tt gn} and {\tt f} with {\tt fn} in the result. A more obvious
rule would be\col\\[2mm]
\begin{minipage}[b]{9cm}
\begin{verbatim}
OPE[a___,b_+c_,d___] := OPE[a,b,d] + OPE[a,c,d]
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:OPElinear2}
\ee
\end{minipage}

However, this rule is much slower in handling sums with more
than two terms. Indeed when there is a sum of $n_1$ ($n_2$) terms in
the first (second)  argument of {\tt OPE}, this rule would have
to be applied $n_1 \times n_2$ times, while {\tt Distribute} handles all
cases in one application. The difference in performance is shown in
fig.\ \ref{fig:linearityPlus}.
\begin{figure}[hbt]
\centering
\portpict{time1}(10,5.2)
\caption[Timings for two different rules to implement bilinearity of
OPEs.]
{Timings for two different rules to implement bilinearity of
OPEs. The timings are for {\tt OPE[A,B]} with {\tt A} of a sum of $10$
terms and {\tt B} a sum of $i$ terms. The plain line is for {\tt
Distribute} \re{v:OPElinear} and the dashed line is for the sum-rule
\re{v:OPElinear2}.
\label{fig:linearityPlus}}
\end{figure}
During the execution of the algorithm, OPEs of sums occur frequently,
hence it is important to select the most efficient rule.
\end{intermezzo}

The other rules that are needed for the {\tt OPE} function
are\col\\[2mm]
\begin{minipage}[b]{9cm} \bv
Literal[OPE[Derivative[i_][A_],B_]]:=
        OPEDerivativeHelpL[A,B,i]
Literal[OPE[A_,Derivative[i_][B_]]]:=
        OPEDerivativeHelpR[A,B,i]
Literal[OPE[A_,NO[B_,C_]]] :=
        CallAndSave[OPECompositeHelpR,A,B,C]
Literal[OPE[NO[A_,B_],C_]] :=
        CallAndSave[OPECompositeHelpL,A,B,C] /;
        Not[SameQ[Head[B],NO]]
Literal[OPE[B_,A_]] :=
        OPECommuteHelp[B,A] /;
        Or[SameQ[Head[B],NO], OPEOrder[A,B]>0]
Literal[OPE[_,_]]= OPEData[{}]
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:OPErules}
\ee
\end{minipage}

Here, {\tt OPEDerivativeHelpL}, {\tt OPE\-De\-rivativeHelpR},
{\tt OPEComposite\-HelpL}, {\tt OPE\-Commute\-Help}
implement eqs.\ \re{eq:RulePolederL}, \re{eq:RulePolederR},
\re{eq:RuleOPEcompL} and \re{eq:RuleOPEcomm}
respectively. {\tt OPE\-Composite\-HelpR}
differentiates between \re{eq:RuleOPEcompRAB} and
\re{eq:RuleOPEcompRBA}. \puzzle{nog niet}
{\tt Call\-And\-Save[f\un,args\un\un]} is defined in \OPEdefs\ to call the
function {\tt f} with arguments {\tt args}, saving the result when
a global switch is set. The last rule declares all non-defined OPEs to be
regular.

Using the {\tt Help} functions has several advantages compared to
inserting their definitions ``in-place''. First, the structure of
the program becomes much clearer. Second, by simply redefining
{\tt OPE\-Composite\-HelpL} and {\tt OPE\-Composite\-HelpR},
we can switch between OPEs and Poisson brackets.

The actual definition of the {\tt Help} functions is of course rather
technical, we will give one example in intermezzo \ref{int:OPEcompHelp}.
However, more important is to understand how the rules for {\tt OPE}
handle an OPE with arbitrarily complicated operators. First of all it is
important to remember the evaluation sequence of \Mathematica. The rules
\re{v:OPErules} are stored in the order they are defined, preceded
by the definitions the user has given for the OPEs of the generators.
Presented an OPE to evaluate, \Mathematica\ starts on top of the list of
rules and applies the first matching rule. Then the evaluation sequence
is restarted. In this way, these rules obviously form an implementation
of the algorithm presented in section \ref{ssct:OPEAlgorithm}.

Note that the rules \re{v:OPErules} avoid making use of the more
complicated function {\tt OPECompositeHelpL} when the first argument of
the OPE is a nested composite, instead {\tt OPE\-Commute\-Help} is used
on the result of {\tt OPEComposite\-HelpR}. This is more efficient in
most cases.

Let us as an example follow \OPEdefs\ in the
computation of a fairly complicated OPE.
\\ \inm{Bosonic[A,B,C,D,E]}
\\ \inm{TracePrint[OPE[NO[A,NO[C',E]],NO[B,D]],}
\\ \contmind{(OPEDerivativeHelpL|OPEDerivativeHelpR|}
\\ \contmind{\ OPECompositeHelpL|OPECompositeHelpR|}
\\ \contmind{\ OPECommuteHelp)[\un\un ]]}
\\ \outm{OPECompositeHelpR[NO[A, NO[C', E]], B, D]}
\\ \contm{\ \ \ \ OPECommuteHelp[NO[A, NO[C', E]], B]}
\\ \contm{\ \ \ \ \ OPECompositeHelpR[B, A, NO[C', E]]}
\\ \contm{\ \ \ \ \ \ \ \ OPECommuteHelp[B, A]}
\\ \contm{\ \ \ \ \ \ \ \ OPECompositeHelpR[B, C', E]}
\\ \contm{\ \ \ \ \ \ \ \ \ \ \ OPEDerivativeHelpR[B, C, 1]}
\\ \contm{\ \ \ \ OPECommuteHelp[NO[A, NO[C', E]], D]}
\\ \contm{\ \ \ \ \ OPECompositeHelpR[D, A, NO[C', E]]}
\\ \contm{\ \ \ \ \ \ \ \ OPECommuteHelp[D, A]}
\\ \contm{\ \ \ \ \ \ \ \ OPECompositeHelpR[D, C', E]}
\\ \contm{\ \ \ \ \ \ \ \ \ \ \ OPEDerivativeHelpR[D, C, 1]}
\\ \contm{\ \ \ \ \ \ \ \ \ \ \ \ OPECommuteHelp[D, C]}
\vspace*{2mm}

\begin{intermezzo}\label{int:OPEcompHelp}%
In this intermezzo, we give an example of the implementation of a {\tt
Help} function, {\tt OPEComposite\-HelpRQ} which corresponds to
\vgl{eq:RuleOPEcompRAB}.
\begin{verbatim}
    OPECompositeHelpRQ[A_,B_,C_] :=
        Block[{q,l,sign = SwapSign[A,B], ABC, AB, AC,
                 maxAB, maxABC, maxq},
            AB = OPE[A,B];
            AC = If[ SameQ[B,C], AB, OPE[A,C]];
            maxAB = MaxPole[AB];
            ABC = Table[
                      OPE[OPEPole[q][AB], C],
                      {q,maxAB}
                  ];
            maxABC = Map[MaxPole, ABC];
            maxq = Max[maxABC + Range[maxAB], MaxPole[AC]];
            maxABC = Max[maxABC,0];
            OPEData[
                Table[
                    OPESimplify[
                        sign * NO[B,OPEPole[q][AC]] +
                        NO[OPEPole[q][AB],C] +
                        Sum[Binomial[q-1,l] *
                           OPEPole[l][ ABC[[q-l]] ] ,
                           {l,Max[1,q-maxAB], Min[q-1, maxABC]}
                        ]
                    ],
                    {q,maxq,1,-1}
                ]
            ]
       ]
\end{verbatim}
Although this is rather lengthy, most of the lines are quite
straightforward. The most important part of the routine is
formed by the statements
bracketed with {\tt OPEData}. Due to the internal representation for
OPEs that is used, the result of the routine is just a table of all
the poles in the OPE -- starting with the highest pole -- with
{\tt OPEData} wrapped around it. Note that {\tt ABC} is defined
such that {\tt OPEPole[l][ ABC[[q-l]] ]} is equal to
$[[AB]_{q-l}C]_l$.

The determination of the order {\tt maxq} of the highest pole in
the OPE requires some explanation. Clearly, the first term
in the \rhs\ of \vgl{eq:RuleOPEcompRAB} gives ${\tt maxq}\geq {\tt
maxAC}$. Furthermore, the terms in the sum over {\tt l} are zero
unless
\[{\tt l} \leq {\tt MaxPole[ABC[[q-l]]]}\]
or also
\[{\tt q-l} \leq {\tt MaxPole[ABC[[l]]]}\]
hence
\[{\tt q} \leq {\tt MaxPole[ABC[[l]]] + l}\]
which is exactly the other boundary on {\tt maxq} which is used.

This shows that the boundaries on {\tt l} and {\tt q} can be computed
without any notion of the dimension of a field. The rule is thus
suitable for any OPA, which is also true for all other {\tt Help}
functions.
\end{intermezzo}
\subsubsection{Rules for derivatives}
The \Mathematica\ symbol {\tt Derivative} has almost no rules
associated with it, except derivatives of the standard functions like
{\tt Sin}. However, we do not need these standard functions, but we do
need linearity of derivatives -- $\dz(A+B)=\dz A + \dz B$ -- which is not
included in the built-in definitions for {\tt Derivative}. We chose not
to use a new symbol for derivatives because of the convenient notation
{\tt A'} for {\tt Derivative[1][A]}. The rules are similar to declaring
linearity of OPEs\col \\[2mm]
\begin{minipage}[b]{9cm}\bv
Derivative[i_][a_Plus] := Map[Derivative[i], a]
Derivative[i_][a_ b_]  :=
        b Derivative[i][a] /; OperatorQ[a]
Derivative[_][0] = 0
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:Derivative}
\ee
\end{minipage}

The first rule handles the $i$-th derivative of a sum. As {\tt
Derivative[i\un]} expects only one argument, a call to {\tt
Distribute} is not necessary. Instead, we map the $i$-th derivative on
all terms in the sum.\\
The second rule handles operators multiplied with
scalars. We never need products of operators in our framework.
\begin{intermezzo}
We comment on the efficiency of the rule for a product given in
\re{v:Derivative}. In \Mathematica\ standard order, {\tt b} follows {\tt
a}. Because {\tt Times} has the attribute {\tt Orderless}, this means
that during pattern matching the {\tt a\un} pattern will be matched
sequentially to every factor in the product, while {\tt b\un} will be the
rest of the product. As we know that the argument of {\tt Derivative[i]}
consists of some scalars times only one operator, we see that for each
scalar {\tt OperatorQ} is called once, until the operator is found. Hence,
it takes maximum $n$ evaluations of {\tt OperatorQ} for a product of $n$
factors. The steps in evaluating {\tt (a b c)'}, where only one of these
is an operator, are\col
\bv
   if OperatorQ[a] then return(b c a')
   elseif OperatorQ[b] then return(a c b')
   elseif OperatorQ[c] then return(a b c')
   endif
\end{verbatim}
This has to be contrasted with the rule\\[2mm]
\begin{minipage}[b]{8cm}
\bv
Derivative[i__][b_ a_]  :=
        a Derivative[i][b] /; OperatorQ[b]
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}[b]{5em}
\be
\label{v:Derivative2}
\ee
\end{minipage}

As {\tt a} is ordered before {\tt b} the steps in evaluating the same
derivative {\tt (a b c)'} are\col
\bv
   if OperatorQ[b c] then
      return( Times[a,
        if OperatorQ[c] then return(b c')
        elseif OperatorQ[b] then return(c b')
        endif
      ])
   elseif OperatorQ[a c] then
      ...
   elseif OperatorQ[a b] then
      ...
   endif
\end{verbatim}
This is clearly more complicated. Moreover, to test {\tt
OperatorQ[b c]} may require two evaluations of {\tt
OperatorQ}. In general, for $n$ factors the worst case would be that
$n(n-1)/2$ tests are needed. The difference in performance is rather
drastic, see fig.\ \ref{fig:linearityTimes} where we include also two
other possibilities using {\tt Not[OperatorQ[a]]}.
\begin{figure}[hbt]
\centering
\portpict{time2}(10,5.2)
\caption[Timings for four different rules to implement extracting of
scalars.]
{Timings for four different rules to implement extracting of
scalars as a function of the number of scalars. The timings are
for the evaluation of {\tt d[A B]} where {\tt A} is a product of $i$
scalars and {\tt B} is an operator.\\
Plain line for {\tt d[a\un b\un] := b d[a] /; Op[a]},\\
dashed line for {\tt d[a\un b\un] := b d[a] /; !Op[b]},\\
dash-dot line for {\tt d[a\un b\un] := a d[b] /; !Op[a]},\\
dotted line for {\tt d[a\un b\un] := a d[b] /; Op[b]}.
\label{fig:linearityTimes}}
\end{figure}
\end{intermezzo}

\subsubsection{Other rules in \OPEdefs}
With the above rules we have already a working package to compute OPEs of
arbitrary nested composites. We also need to bring composites in a
standard form. This is done by {\tt NO}. The rules attached to {\tt NO}
are entirely analogous to those for {\tt OPE}.

One also needs some extra functions like {\tt OPESimplify}. We do not
give their implementation here. One can always consult the source of
\OPEdefs\ for further information.

One major ingredient of \OPEdefs\ is not yet discussed\col\
{\tt OPEPole}. For the operation of the above rules to work,
only a very simple definition for the {\tt OPE\-Pole} of an {\tt
OPEData} is needed\col
\bv
OPEPole[n_][OPEData[A_]] :=
        If [1 <= n <= Length[A], A[[-n]], 0]
\end{verbatim}
\OPEdefs\ allows also a syntax \verb&OPEPole[i][A,B]&.
This can be used to compute only one pole of the complete OPE, avoiding
the computation of all the other poles. Currently, the rules for {\tt
OPEPole} are completely independent of those to compute a full OPE. This
is because when computing \eg {\tt OPE[A, NO[B, C]]} (see {\tt
OPECompositeHelpR} above), it is more efficient to compute {\tt OPE[A,
B]} and {\tt OPE[A, C]} once and save these results. It is not a good
programming practice to keep two separate sets of rules computing
essentially the same thing, but efficiency seemed to be more important
at this point.

\subsection{Performance}
In \cite{OPEdefs2.0}, a Wakimoto \cite{Wakimoto} realisation for the \KA\
$\widehat{B_2}$ with level $k$ using $2$ free scalars and
$4$ (bosonic) $\beta$, $\gamma$ systems
was constructed. Describing this realisation would lead us too far here,
but to give an idea of the complexity of the calculation,
the total number of composites in the realisation is rougly sixty, and
composites of up to four free fields are used.

In table \ref{table:CPU}, we tabulate CPU times for
computing an OPE of two of the currents, and the Sugawara tensor for this
realisation. The first time given in the table is the time for evaluating
the statement after loading the package and defining the realisation.  The
time between brackets is measured when the statement is repeated. The
second execution is much faster because \OPEdefs\ stores some of the
OPEs with composites.
Note that version 2.0 of \Mathematica\ is roughly 1.4 times slower than
version 1.2.
\begin{table}[hbt]
\caption{CPU time for the computation of the OPE of the currents
corresponding to the positive simple root of $\widehat{B_2}$ (statement
9) and the computation of the Sugawara tensor (statement 11) (see Ref.
\protect\cite{OPEdefs2.0}) for \Mathematica\ running on a PC 386 (25
Mhz).
\label{table:CPU} }
\vspace{.1 cm}
\begin{center}
\begin{tabular}{|l||c|c|c|}
\hline
                      &             &              & \\*[-.2cm]
{\Mathematica}-version &    1.2     &  1.2         & 2.0 \\
{\OPEdefs}-version     &    2.0     &  3.1         & 3.1 \\
                      &             &              & \\*[-.2cm]
\hline
                      &             &              & \\*[-.2cm]
{\tt In[9]}
                      &23.5 (4.5) s & 14.9 (2.8) s & 19.3 (3.8) s \\
{\tt In[11]}
                      &43.2 (11.6) s& 31.3 (9.4) s & 40.7 (12.1) s\\*[.2cm]
\hline
\end{tabular}
\end{center}
\end{table}

\section{User's Guide\label{sct:OPEdefsGuide}}
\setcounter{mathline}{0}
This section is intended as a user's guide to the package \OPEdefs\
3.1. Explicit examples are given for most operations. Note that
\OPEdefs\ 3.1 requires \Mathematica\ 1.2 or later.

We introduce some special notations. Input for and output from
\Mathematica\ is written in {\tt typeset} font. Input lines are preceded
by ``{\sl In[n] :=}'', and corresponding output statements by ``{\sl
Out[n] =}'', as in \Mathematica.

As \OPEdefs\ is implemented as a \Mathematica\ package, it has to be
loaded {\em before} any of its global symbols is used. Loading the
package a second time will clear all previous definitions of operators
and OPEs, as well as all stored intermediate results. Assuming that the
package is located in the {\sl Mathematica}-path, \eg\ in your current
directory, issue\col
\\[2mm] \inm{<<OPEdefs.m}\\[2mm]
After loading \OPEdefs\ into \Mathematica, help for all the global
symbols is provided using the standard help-mechanism, \eg {\tt ?OPE}.

Now, you need to declare the operators that will be used. If you
want to define bosonic operators {\tt T} and {\tt J[i]} (any index
could be used), and fermionic operators {\tt psi[i]}, the
corresponding statements are\col
\\[2mm] \inm{Bosonic[T, J[i\un]]}
\\ \inm{Fermionic[psi[i\un]]}\\[2mm]
The order of the declarations fixes also the
ordering of operators used by the program\col
\be
\mbox{{\tt T < J[1]' < J[1] < J[2] < J[i] < psi[1] < ...
}} \label{eq:OPEdefsOrder}
\ee
By default, derivatives of an operator are considered ``smaller'' than
the operator itself. This can be reversed using the global options {\tt
NOOrdering} (see  below).

Finally, the non-regular OPEs between the basic operators
have to be given. An OPE can be specified in two different ways.\\
The first way is by listing the operators that occur at the poles, the
first operator in the list is the one at the highest non-zero pole,
the last operator has to be the one at the first order pole, \eg
\col
\\[2mm] \inm{OPE[T, T] = MakeOPE[\lb c/2 One, 0, 2T, T' \rb];}\\[2mm]
Note the operator {\tt One} which specifies the unit-operator.\\
The second way is by giving the OPE as a Laurent series expansion,
adding the symbol {\tt Ord} which specifies the (implicit) arguments of the
operators for which the OPE is defined\footnote{The first time
you use this syntax, you may notice an unexpected delay. This is
because \Mathematica\ is loading the {\tt Series} package.}.
The arguments for the operators can be any \Mathematica\ expression.

\noindent{\bf Warning}\col\ it is important that the operators occuring as
arguments of {\tt OPE} in a definition should be given in standard
order (\ref{eq:OPEdefsOrder}), otherwise wrong results will be
generated.

The following statements define a $\widehat{SU(2)}_k$-\KA\col
\\[2mm] \inm{OPE[J[i\un],J[i\un]] :=}
\\ \contm{ MakeOPE[-k/2 (z-w)\ha-2+ Ord[z,w,0]]}
\\ \inm{OPE[J[1],J[2]] = }
\\ \contm{ MakeOPE[J[3][w](z-w)\ha-1 +Ord[z,w,0]];}
\\ \inm{OPE[J[2],J[3]] =}
\\ \contm{  MakeOPE[J[1][w](z-w)\ha-1 + Ord[z,w,0]];}
\\ \inm{OPE[J[1],J[3]] =}
\\ \contm{  MakeOPE[-J[2][w](z-w)\ha-1 + Ord[z,w,0]];}\\[2mm]
In fact, with the above definitions, one has to use always
the explicit indices $1,2,3$ for the currents $J$. If we would compute
an OPE with current {\tt J[i]} where the index $i$ is not $1,2$ or $3$,
wrong results will be given. One can circumvent this peculiarity by
reformulating the definitions.

A normal ordered product $[ A B ]_0$ is entered in the form {\tt
NO[A,B]}. Multiple composites can be entered using only one {\tt
NO} head, \eg {\tt NO[A,B,C]}. This input is effectively translated
into {\tt NO[A, NO[B, C]]}. All output is normal ordered with the same
convention, \ie from right to left (input can be in any order). Also,
the operators in composites will always be
ordered according to the standard order (\ref{eq:OPEdefsOrder}).\\
As an example, we can define the Sugawara energy-momentum tensor for
$\widehat{SU(2)}_k$. The \Mathematica\ output of an OPE is a list of the
operators at the poles.
\\[2mm] \inm{Ts = -1/(k+2)(NO[J[1],J[1]]+}
\\ \contmind{NO[J[2],J[2]]+NO[J[3],J[3]]);}
\\ \inm{OPESimplify[OPE[Ts, J[1]]]}
\\ \outm{<< 2|| J[1] ||1|| J[1]' >>}
\setcounter{mathref}{\value{mathline}}
\vspace*{2mm}

\noindent{\bf Warning}\col\ when computing OPEs with composites, or when
reordering composites, \OPEdefs\ remembers by default some
intermediate results. Thus, it is dangerous to change the definition of
the basic OPEs after some calculations have been performed. For example,
consider a constant $a$ in an OPE. If calculations are performed after
assigning a value to $a$, the intermediate results are stored with this
value. Changing $a$ afterwards will give wrong results.

The other globally defined functions available from the package
are\col
\begin{itemize}
\item \verb&OPEOperator[operator_, parity_]& provides a more general way to
declare an operator than {\tt Bosonic} and {\tt Fermionic}. The second
argument is the parity of the operator such that $(-1)^{\tt parity}$ is
$+1$ for a boson, and $-1$ for a fermion. It can be a symbolic constant.
This is mainly useful for declaring a $bc$-system of unspecified parity, or a
\KA\ based on a super-Lie algebra. In such cases, the operator can contain a
named pattern\col
\\ \inm{OPEOperator[J[i\un],parity[i]]}\\
If one wants to declare more operators, one can group each operator and its
parity in a list\col
\\ \inm{OPEOperator[\{b[i\un],parity[i]\},\{c[i\un],parity[i]\}]}\\
See also {\tt SetOPEOptions[ParityMethod, \un]}.
\item \verb&OPEPole[n_][ope_]& gets a single pole term of an OPE\col
\\ \inm{OPEPole[2][Out[\themathref]]}
\\ \outm{J[1]}\\
\verb&OPEPole[n_][A_,B_]& can also be used to compute only one pole term of
an OPE\col
\\ \inm{Factor[OPEPole[4][Ts, Ts]]}
\\ \outm{(3 k One)/(2 (2 + k))}\\
{\tt OPEPole} can also give terms in the regular part of the OPE\col
\\ \inm{OPEPole[-1][T, T]}
\\ \outm{NO[T', T]}
\item \verb&MaxPole[ope_]& gives the order of the highest pole in the OPE.
\item \verb&OPEParity[A]& returns an even (odd) integer of $A$ is
bosonic (fermionic).
\item \verb&OPESimplify[ope_, function_]& ``collects'' all terms in
{\tt ope} with the same operator and applies {\tt function} on the
coefficients. When no second argument is given, the coefficients
are {\tt Expand}ed.
\\[2mm] \inm{OPESimplify[OPE[J[1], NO[J[2], J[1]]]]}
\\ \outm{<< 2|| (1 - k/2) J[2] ||1||}
\\ \contm{\ \ \ NO[J[1], J[3]] + J[2]' >>}\\[2mm]
\verb&OPESimplify[pole_, function_]& does the same simplifications
on sums of operators.
\item \verb&OPEMap[function_, ope_]& maps {\tt function} to all poles of
{\tt ope}.
\item \verb&GetCoefficients[expr_]& returns a list of all
coefficients of operators in {\tt expr} which can be (a list of) OPEs
or poles.
\item \verb&OPEJacobi[op1_,op2_,op3_]& computes the
Jaco\-bi-iden\-ti\-ties \re{eq:OPEJacRAB} for the singular part of
the OPEs of the three arguments. Due to the nature of
\vgl{eq:OPEJacRAB}, the computing time will be smallest (in most
cases) when {\tt op1} $\leq$ {\tt op2} $\leq$ {\tt op3} in the order
\re{eq:OPEdefsOrder}. In nonlinear algebras, the computation
will use rules for OPEs of composites which assume that the Jacobi 
identities hold. This means that the result of {\tt OPEJacobi} gives
only necessary conditions. In some cases, different orderings of
{\tt op1-op3} have to be tried to find all conditions.\\
The result of {\tt OPEJacobi} is a double list of operators. It is
generated by\\
\hspace*{4em}{\tt
Table[OPEPole[n][A,OPEPole[m][B,C]] +}\\ \hspace*{8em}{\tt {\sl
corrections},\  \{m, maxm\},\{n,maxn\}]}\\
All elements of the list
should be zero up to null operators for the OPA to be associative.
\item \verb&Delta[i_,j_]& is the Kronecker delta symbol
$\delta_{ij}$.
\item \verb&ClearOPESavedValues[]& clears all stored
intermediate results, but not the definition of the operators and
their OPEs. To clear everything, reload the package.
\item \verb&OPEToSeries[ope_]& converts
an OPE to a Laurent series expansion in {\tt z} and {\tt w}. The
arguments can be set to {\tt x} and {\tt y} with\col
\\ \inm{SetOPEOptions[SeriesArguments, \{x, y\}]}
\item \verb&TeXForm[ope_]& gives \TeX output for
an OPE. The same arguments are used as in \verb&OPEToSeries&.
\item \verb&OPESave[filename_]& (with {\tt filename} a string between
double quotes) saves the intermediate results that \OPEdefs\ remembers
to file (see the option {\tt OPESaving} below).
\item {\tt SetOPEOptions} is a function to set the global options of
the package. The current options are\col
\begin{itemize}
\item \verb&SetOPEOptions[SeriesArguments, {arg1_,
arg2_}]& : sets arguments to be used by {\tt TeXForm} and {\tt
OPEToSeries}. One can use any Mathematica expression for {\tt arg1}
and {\tt arg2}.
\item \verb&SetOPEOptions[NOOrdering, n_]& :
if {\tt n} is negative, order higher derivatives to the left
(default), if {\tt n} is positive, order them to the right.
\item \verb&SetOPEOptions[ParityMethod, 0|1]& : makes it possible to use
operators of an unspecified parity. When the second argument is $0$
(default), all operators have to be declared to be bosonic or
fermionic. When the argument is $1$, {\tt OPEOperator} can be used with
a symbolic parity.
Note that in this case, powers of $-1$ are used to compute signs, which is
slightly slower than the boolean function which is used by the first
method.\\
This option is not normally needed as the use of {\tt
OPEOperator} with a non-integer second argument sets this option
automatically.
\item \verb&SetOPEOptions[OPESaving, boolean_]& :
if {\tt boolean} evaluates to {\tt True} (default), \OPEdefs\
stores the intermediate results when computing OPEs of composites and
when reordering composites. This option is useful if \Mathematica\ runs
short of memory in a large calculation, or when computing with dummy
indices\footnote{No mechanism to use dummy indices is built-in in
\OPEdefs. I wrote a separate package {\sl Dummies} to handle
this.}.
\item \verb&SetOPEOptions[OPEMethod, method_]& :
with the parameter {\tt method} equal to {\tt Quan\-tum\-OPEs} enables
normal OPE computations (default setting), while {\tt Classi\-cal\-OPEs}
enables Poisson bracket computations. Using this option implicitly calls
\verb&ClearOPESavedValues[]&.
\end{itemize}
\end{itemize}
\section{Example : The conformal anomaly in superstring
theory\label{eq:OPEexample}}
\setcounter{mathline}{0}
We consider only one free boson field $X$ and one free fermion
field $\psi$ because additional free fields will have exactly the same OPEs
and commute with each other. We denote $\dz X$ with {\tt J} and $\psi$
with {\tt psi} (we normalise them such that they have a $+1$ in their
OPEs \re{eq:dzXdzXOPE} and \re{eq:psipsiOPE}). The ghosts are a
fermionic $b$, $c$ system (operators {\tt b,c}) and a bosonic $\beta$,
$\gamma$ system (operators {\tt B,G}) (normalised such that
${}_iA^j=(-1)^{i+1} {}_i\delta^j$ in \vgl{eq:OPEbc}). $b$ has conformal
dimension $2$ and $\beta$ has $3/2$. It is now a
trivial task to compute the conformal anomaly\col
\\[2mm] \inm{<<OPEdefs.m}
\\ \inm{Bosonic[J,B,G]; Fermionic[b,c,psi];}
\\ \contm{OPE[J,J] = MakeOPE[\{One, 0\}];}
\\ \contm{OPE[psi,psi] = MakeOPE[\{One\}];}
\\ \contm{OPE[b,c] = MakeOPE[\{One\}];}
\\ \contm{OPE[B,G] = MakeOPE[\{One\}];}
\\ \contm{Tb = 1/2 NO[J,J]; Tf = -1/2 NO[psi,psi'];}
\\ \contm{Tbc = -2 NO[b,c'] - NO[b',c];}
\\ \contm{TBG = 3/2 NO[B,G'] + 1/2 NO[B',G];}
\\[2mm] \inm{OPESimplify[OPE[Tb,Tb]]}
\\ \outm{<< 4|| One/2 ||3|| 0 ||2|| NO[J, J] ||1||}
\\ \contmind{ NO[J', J] >>}
\\ \inm{OPESimplify[OPE[Tf,Tf]]}
\\ \outm{<< 4|| One/4 ||3|| 0 ||2|| NO[psi', psi] ||1||}
\\ \contmind{ NO[psi'', psi]/2>>}
\\ \inm{OPESimplify[OPE[Tbc,Tbc]]}
\\ \outm{<< 4|| -13 One ||3|| 0 ||2||}
\\ \contmind{ -4 NO[b, c'] - 2 NO[b', c]||1||}
\\ \contmind{ -2 NO[b, c''] - 3 NO[b', c'] - NO[b'', c] >>}
\\ \inm{OPESimplify[OPE[TBG,TBG] - MakeOPE[\{2 TBG,TBG'\}]}
\\ \outm{<< 4|| 11 One/2 ||3|| 0 ||2|| 0 ||1|| 0 >>} \\[2mm]
We see that each bosonic (fermionic) field will contribute a
central charge $1$ ($1/2$) to the total central charge of the theory.
The $b$, $c$ system contributes $-26$, and the $\beta$, $\gamma$
system $11$. This gives the well known relation for the critical
dimensions of the bosonic string $D_b - 26 = 0$ and the superstring $3/2
D_s -26 + 11 = 0$.  Moreover, we can easily verify that the {\emt}s obey
the Virasoro algebra .

The reader without experience in CFT is invited at this point to take
out some time and compute the OPE for $T_{BG}$, for instance, by hand.
Although this computation is rather trivial with \OPEdefs, the same
calculation was attempted in \cite{SUPERCALC} using the mode--algebra.
There it proved not to be possible to compute the Virasoro algebra
automatically due to difficulties with the infinite sums in the normal
ordered products.

\section{Future developments\label{sct:OPEfuture}}

A first extension would be to add the possibility to specify a range of
poles one wants.  This would unify the current implementations of {\tt OPE}
and {\tt OPEPole}.  The main difficulty in programming this, is that
when saving intermediate results, one has to see which poles are already
computed and which not.

The objective in writing \OPEdefs, was to make a package available
which is as general as possible. One could write specialised extensions
which would outperform \OPEdefs. For instance, the restriction to free
fields would be very useful. Also \WA s, a conformal OPA generated by
quasiprimaries, form a preferred subclass of the OPAs. In this case it
would be advantageous to compute in a basis of quasiprimaries, see
section \ref{sct:globalConformalRestrictions}. A package under
development \cite{OPEconf} collects some formulas for working with
(quasi)primaries in a \WA, but it relies on \OPEdefs\ for computing OPEs.

The main restriction of \OPEdefs\ is of course the requirement of poles of
integer order.  In particular, vertex operators are widely used in
conformal field theory.  Here the order of the powers in the
generalised Laurent expansion remain integer spaced. This should make it
possible to extend \OPEdefs\ to handle this case. However, the notion of the
singular and regular part of an OPE is not so important when using vertex
operators as in other cases. Indeed, the normal ordered product of
two vertex operators should not be defined as the zeroth order pole in
their OPE. This makes it desirable to use a data representation
which keeps the information to generate any ``pole", but stores already
computed results\footnote{This is a well known concept called ``streams"
in the symbolic manipulation program {\sl Axiom}, where \eg Taylor series
are stored in this way.}. This would make it possible to work with vertex
operators $V_a$ with a symbolic weight $a$, and not only with fixed
numbers. One could then also define an OPE between fields of
(non-numeric) dimension $h_1,h_2$, \eg with the unit operator at the pole of
order $h_1+h_2$. Finally, the extension to nonmeromorphic OPEs would allow to
treat parafermions \cite{parafermion1,parafermion2,parafermion3}.

We are currently working at a version for computing with super OPEs in
$N=2$ superfields \cite{N2SOPEdefs}. Extension to arbitrary $N$ will not
give great difficulties.

Finally, it would be convenient to be able to use dummy indices in
\OPEdefs. I wrote a separate package {\sl Dummies} to handle this.
However, this package uses only a rudimentary algorithm to simplify
expressions with dummy indices. There seems to exist no algorithm (except
exhaustive enumeration) to do this simplification for the case at hand.
The main difficulty is that correction terms are needed when
interchanging operators in a normal ordered product.


\section{Other packages}
As shown in section \ref{sct:modeAlgebra}, an other approach to the
same problem would be to use modes. An attempt to compute
commutators of the modes of normal ordered operators in {\sc reduce}
\cite{SUPERCALC} was not completely successful due to
difficulties in the reshuffling of the indices in infinite sums. It
should be possible to avoid this by constructing the
formulas for commutation with a mode of a composite operator by
looking at the corresponding formula for OPEs. Apparently, this
is done in a package by L. Romans, which is not published, but
acknowledged by a few authors. The mode-approach has probably the
advantage that contributions of the tower of derivatives of a
quasiprimary operator (see section
\ref{sct:globalConformalRestrictions}) are summed in advance. However,
this definition \re{eq:modedef} which makes the formulas for computing
with modes convenient is restricted to a conformal OPA.

Related to mode calculations is the approach taken by H.\
Kausch (using another symbolic manipulation program {\sl Maple}). He uses
the equivalence between conformal fields $\Phi(x)$ and states
$\Phi(0) |0\!>$ discussed in \cite{Goddard}. As this reference restricts
itself to bosonic fields of integer dimension, and fermionic fields of
half-integer dimension, we suspect that Kausch's program will not be
able to handle algebras outside this framework, like 
\cite{PolyBosSusy,mic,filipzbigquadratic}, while 
they present no problems for \OPEdefs.
However, Kausch did not publish his work.

Finally, A.\ Fujitsu recently developed a package, {\sl ope.math}, in
\Mathematica\ for computing OPEs of free fields \cite{Fujitsu}. It is
able to treat vertex operators at non-integer poles (this is not possible
in \OPEdefs). However, one cannot compute OPEs in a \WA\ or a \KA, except
by working in a realisation with free fields. The current version is much
slower than \OPEdefs\ for calculations without vertex operators. {\sl
ope.math} uses a fixed notation for all fields, \eg the $i$-th derivative
of a complex boson (part of a bosonic $b, c$-system, see section
\ref{sct:FreeFields}) has to be called {\tt be[i, j, z]} where $j$ is an
index and $z$ the coordinate. The package is currently not able to work
with an unspecified number of fields, \ie {\tt ope[be[0,i, z], ga[0,j,
w]]} returns zero.

%12345678901234567890123456789012345678901234567890123456789012345678901234567890
%J_\zb = \k/2 \db g g^{-1}
\chapter{Renormalisation factors in $\Ww$--Gravity\label{chp:renormalisations}}
\mychapter
In this chapter, we will study induced gravity theories in the light-cone
gauge.  For \WA s which can be realised in terms of a constrained \WZW, we
prove in section \ref{sct:RNallorder} that the effective action can be
computed by simply inserting renormalisation factors in the classical
result. We give explicit expressions for these factors to all orders in the
coupling constant.

As an example, the supergravity theories based on the $N$-extended $\soN$
superconformal algebras will be presented in section \ref{sct:RNsoN}.

In section \ref{sct:RNsemiclass} the renormalisation factors for the linear
superconformal algebras are computed in a semiclassical approximation.
These results are compared with the all--order expressions of
\ref{sct:RNsoN}.

%Appendix \ref{sct:RNWZWfactors} presents a derivation of the
%renormalisation factors in a gauged \WZW.

The discussion relies on the results of chapter \ref{chp:WZNW}. Conventions
on \WZW s are given in section \ref{sct:WZW}.

This chapter is based on \cite{us1}. However, several topics have been
expanded, especially in section \ref{sct:RNallorder}.

\section{Introduction}
In two dimensions, the Einstein-Hilbert action for the metric $g^{ij}$\col
\be
\int d^2x\ \sqrt{g} R\,,
\ee
where $R$ is the Riemann curvature scalar, is a topological invariant
and hence trivial. This means that any action where the matter fields are
covariantly coupled to the metric can be regarded as an action for the
matter coupled to gravity. Consider as an example for the matter system a
scalar boson\col
\be
S[X,g] = -{1\over 4\pi}\int dx^2\,\sqrt{g} g^{ij}\nabla_iX(x)\nabla_jX(x)\,,
\ee
where $\nabla_i$ denotes a covariant derivative. In the light-cone gauge
$ds^2=2dzd\zb+2\mu d\zb d\zb$, this action reduces to the free field action
with a flat metric \re{eq:actionFreeFieldg} $S_s(X,\delta^{ij})$ plus a
coupling of the metric to the holomorphic component of the \emt\col
\be
S[X,g] = S_s(X,\delta^{ij}) + {1\over \pi}\int \mu T\,.
\ee
By integrating out the matter field $X$, we obtain an induced action for
gravity in two dimensions. We see that the induced action
is equal to the generating functional $\Gamma$ \re{eq:indActdef} of the
Virasoro algebra. Similarly, when the matter system specifies a conformal
invariant system, the only remnant of the matter system in the induced
action is the central charge of $T$.

Consider now a matter system, where we denote matter fields collectively by
$\varphi$, with a set of holomorphic global symmetries generated by
$T^i[\varphi]$. We will assume that the matter system is conformally
invariant, and the $T^i[\varphi]$ generate a \WA. The symmetries can be
gauged to make them local. The gauge fields $\mu_i$ correspond to the
remnant of the metric and its generalisations for the higher dimension
fields\puzzle{}
in the light-cone gauge. The induced action for
$\Ww$--gravity coupled to the matter $\varphi$ in the light-cone
gauge is defined by\col
\bea
\exp\left( -\G_{\rm ind}[\m]\right)=\int[d\varphi]\exp\left(
    -S[\varphi]-\frac 1 \pi\int \m_i T^i[\varphi ]\right).
\label{eq:RNindActGravity}
\eea
The gauge fields $\m_i$ are called generalised Beltrami differentials.
When after quantisation, the $T^i$ form a quantum \WA, the induced action
\re{eq:RNindActGravity} is equal to the generating functional $\Gamma$
\re{eq:indActdef} of the \WA.

For $\Ww_3$--gravity, it was shown in \cite{ssvna} that the induced
action \re{eq:RNindActGravity} $\G_{\rm ind}$ can be expanded in a power
series of the inverse central charge $1/c$\col
\be
\G[\mu]=\sum_{i\geq 0}c^{1-i}\G^{(i)}[\mu] \, .
\label{eq:RNcexp}
\ee
A similar expression will be true for other nonlinear \WA s for which a
classical limit $c\rightarrow\infty$ exists, while for most linear algebras
only $\G^{(0)}$ is non-zero\puzzle{alle centr.ext prop to c, Vir+KM}.
The subleading terms in $1/c$ in
\vgl{eq:RNcexp} arise from a proper treatment of the composite terms in
the OPEs. In \cite{ZFacW3ssvnb}, an explicit form for the classical term
$\G^{(0)}$ of $\Ww_3$ was obtained through the classical reduction of an
$Sl(3, \bf{R})$ \WZW. The higher order terms are more difficult to
compute and do not bear any straightforward relation to $\G^{(0)}$.

This complicated behaviour seems to simplify drastically when going to
the effective theory -- \ie quantising the metric and other gauge fields
of the $\Ww$--gravity theory. The generating functional $W$ of the
connected Green's functions, which upon a Legendre transform becomes the
effective action\footnote{We will use the term 'effective action' also
for the generating functional of the connected Green's functions. We
trust the symbol $W$ instead of $\Gamma$ is enough to avoid confusion.}
is defined by\col \be
\exp\left(-W[\check{T}]\right) = \int [d\mu] \,
  \exp\left(-\G[\mu]+\frac{1}{\pi} \int \mu_i \check{T}^i\right)
\, . \label{eq:RNpi2}
\ee
On the other hand, the Legendre transform
$W^{(0)}[\check{T}]$ of $\G^{(0)}[\mu]$ is defined by\col
\be
W^{(0)}[\check{T}]=\min_{\mu}\left( \G^{(0)}[\mu]-
\frac{1}{\pi}\int \mu \check{T}\right)\,.
\label{eq:RNW0def}
\ee
$W^{(0)}[\check{T}]$ satisfies a functional equation which is
given by the classical (\ie large $c$) limit of the Ward identity for
$\G[\mu]$ by interchanging $\{\check{T}, \delta\G^{(0)}/\delta\mu\}$ and
$\{\delta W^{(0)}/\delta\check{T},\mu\}$.

We will prove in this chapter, for \WA s arising from a Drinfeld-Sokolov
reduction, that the relation between these functionals is\col
\be
W[\check{T}] = Z_W\, W^{(0)} \left[ Z^{(\check{T})} \check{T} \right]
\,, \label{eq:RNexact}
\ee
where the renormalisation factors are functions of the central charge
$c$. This was a long-existing conjecture in the literature. It was proven
in \cite{PolyWardT,KPZ} for the Virasoro algebra, and conjectured for
$\Ww_3$ in \cite{ZFacW3ssvnb,ssvnc}. The conjecture was based on a
computation of the first order quantum corrections to $W^{(0)}$ which
showed that the corrections split into two parts\col\ one part
contributes to the multiplicative renormalisations of $W^{(0)}$ while the
other cancels $\G^{(1)}$.

Eq.\ \re{eq:RNexact} implies that the one particle irreducible, or
effective action is simply given by\col
\be
\G_{\rm eff}[\mu] =
Z_W \, \G^{(0)} \left[ \frac{1}{ k_c Z_{(\check{T})}} \mu\right]\,.
\ee

\begin{intermezzo}
The resemblance of the definition \re{eq:RNpi2} to a Fourier transform,
leads to the idea to prove \vgl{eq:RNexact} by considering the
functional equations arising from Ward identities. Unfortunately, this does
not work. We will as an example consider the Virasoro algebra, for which
the Ward identity \re{eq:WardIdTGamma} is\col
\be
-{c\over 12\pi}\dz^3 \mu\ Z[\mu]\ =\
\db \ddt{Z}{\mu} - \mu \dz \ddt{Z}{\mu} - 2 \dz \mu \ddt{Z}{\mu}\,.
\label{eq:RNWardIdTZ}
\ee
We multiply \vgl{eq:RNWardIdTZ} with $\exp\left({1\over\pi}\mu
\check{T}\right)$ and integrate over $\mu$. Assuming we can use functional
partial integration and discard boundary terms, we find\col
\bea
-{c\over 12}\dz^3 \ddt{F}{\check{T}}\!\!&=&\!\!
   -\db \check{T}\ F[\check{T}] -
   \pi\left(\dd{\check{T}} \dz + 2 \dz \dd{\check{T}}\right)
       \int [d\mu]\ \exp\left(\int{1\over\pi}\mu\check{T}\right)\ddt{Z}{\mu}
   \nonu
&=&\!\! -\db \check{T}\ F[\check{T}] +
   \left(\dd{\check{T}} \dz + 2 \dz \dd{\check{T}}\right)
       \left(\check{T} F[\check{T}]\right)
\,, \label{eq:RNrenormtry}
\eea
where we defined\col
\be
F[\check{T}] \equiv \exp\left(-W[\check{T}]\right)\,.
\ee
Obviously, \vgl{eq:RNrenormtry} contains infinite $\delta^{(2)}(0)$ terms.
This is because the Ward identity \re{eq:RNWardIdTZ} suffers short-distance
singularities when $\mu$ is a quantum field.
\end{intermezzo}

The conjecture \re{eq:RNexact} was elegantly proven for the
case of $\Ww_3$ in \cite{ZFacW3dbg} through the use of a quantum
Drinfeld-Sokolov reduction. This chapter pre\-sents the generalisation
for Drinfeld-Sokolov reductions of any \KA\ \cite{us1}. The method is based
on observations in \cite{aleks,quanHRBO}. Using the definition of the
induced action \vgl{eq:RNindActGravity}, the generating functional of its
connected Green's functions \re{eq:RNpi2} becomes\col
\bea
\exp\left( -W[\check{T}]\right)=
\int [d\varphi]\ \delta(T[\varphi ] -\check{T})
     \exp\left(-S[\varphi]\right)\,.
\eea
Evaluating this functional integral is in general impossible. Even when
the number of degrees of freedom in the matter fields is the same as the
number of generators $T$, it involves the computation of a usually very
complicated Jacobian. However, if one takes for the matter system a
gauged \WZW, we can actually compute the Jacobian and as a result obtain
an all--order expression for the effective action.

\section{All--order results\label{sct:RNallorder}}
Consider a \WZW\ with action $\k S^-[g]$. The holomorphic symmetry
currents form a \KA\ $\hat{g}$. In the previous chapter, we proved that
by imposing certain constraints on the currents of $\hat{g}$, we find a
realisation of a \WA. Moreover, we obtained an all-order expression for
the induced action of the corresponding $\Ww$--gravity theory. To make
the transition to the effective theory, we first calculate the induced
action in the highest weight gauge.

The induced action $\Gamma$ \re{eq:RNindActGravity} for the $\Ww$--gravity
is given by\col
\bea
\exp \left(-\Gamma[\mu]\right)&=&\int
[\d g\, g^{-1}][d\t][d A_\zb]\left( \mbox{Vol}\left( \P_+\bar{g}
\right) \right)^{-1}\nonu
&&\qquad\qquad
\exp\left( -{\cal S}_1 -\frac{1}{4\pi xy} \int \str \left(\mu\ct\right)\right)\,,
\label{eq:RNindactstep}
\eea
where ${\cal S}_1$ is defined in \re{eq:RNBactiongen}. The generators
of the \WA\ $\ct^{(j, \alpha_j)}$ are assembled in a matrix\col
\be
\ct\equiv \ct^{(j, \alpha_j)}\ {}_{(jj, \alpha_j)}t\,,
\ee
where the matrices ${}_{(jj, \alpha_j)}t$ form a representation of
the (super) Lie algebra $\bar{g}$ (see appendix \ref{app:super}).
A similar matrix $\in\KER{-}$, exists for the sources $\mu^{(j,\alpha_j)}$.

We will define the path integral in \vgl{eq:RNindactstep} by performing a
gauge fixing in the BV formalism, where we use the quantum extended
action found in the previous chapter \re{eq:RNSqbv}.
\puzzle{opmerking regularistie}
Going to the highest
weight gauge $\P_{\geq 0}J_z\in\KER{+}$ and $\t\!=\!0$ is accomplished by
a canonical transformation which interchanges fields and antifields for
$\t$ and $(\P_{+}\!-\!\PHW )J_z$ and dropping all (new)
antifields. To obtain the induced action, we integrate the resulting
gauge-fixed action over the new fields. For clarity, we do not rename the
fields involved in the transformation and we insert explicit delta
functions in \vgl{eq:RNSqbv}\col
\bea
\leqn{\exp \left(-\G[\mu]\right)=\int
[\d g\, g^{-1}][d\t^*][d A_\zb][dJ^*_z]}
&&\delta\left((\P_{\geq 0}-\PHW )J_z\right)\
  \delta\left(\PLW J^*_z\right)\ \delta\left(\P_+J^*_z\right)\
  \delta(\t)\ \delta\left(A^*_\zb\right)\nonu
&&\exp \Biggl(
-\k S^-[g]- \frac{1}{\pi x} \int \str\left(
     A_\zb\left( J_z -\frac \k 2 e_-\right)\right)-
   \frac{\k}{2\pi x}\int \str \left( \t^*c \right)\nonu
 &&\ \ \  - \frac{\k}{2\pi x}\int \str \left(
     J^*_z\left(\frac \k 2 \del c + [c,J_z] \right)\right)
     -{1\over 4\pi xy}\int \str\left( \mu\, \hat{\ct}\right)
\Biggr)\,.
\label{eq:RNindactstep1}
\eea
The matrix $\hat\ct$ in \vgl{eq:RNindactstep1} was determined in the
previous chapter, see section \ref{sct:RNSBVq}. Important at this moment
is that $\hat\ct$ does not contain the fields $A_\zb, \t^*$ and $J^*_z$. This
enables us to prove the following lemma.
\begin{lemma}
The functional integration over $A_{\zb}$, $\t^*$ and $J^*_z$ in
\vgl{eq:RNindactstep1} gives two additional delta functions\col
\[
\delta\left(\P_-( J_z -\frac \k 2 e_-)\right)\delta(c)\,.
\]
\end{lemma}
\begin{proof}
It is clear that the functional integration over $A_{\zb}$ and $\t^*$
gives\col
\[ \delta\left(\P_-( J_z -\frac \k 2 e_-)\right)\
   \delta\left(\P_{1/2}c\right)\,.
\]
We now integrate over $\P_{-m} J^*_z$ in increasing order of $m$. We
show by recursion that each step in $m$ gives an additional
$\delta(\P_{m+1}c)$, relying on the fact that $\P_{<(m+1)}c$ can be
set to zero due to the previous steps. The recursion assumption
is valid for $m=0$.

Consider the part of the exponent in \vgl{eq:RNindactstep1} which contains
the anticurrents\col
\[
\int\str \left(\P_{-m}J^*_z\left(\frac \k 2 \del \P_m c +
\sum_{m'}[\P_{m'}c,\P_{m-m'}J_z] \right)\right)\,.
\]
The recursion assumption implies that the $\del \P_m c$ term can be
set to zero, and restricts the sum to $m'\geq m+1$.
However, because of the delta functions of the currents, the terms in
$\P_{m-m'}J_z$ are either highest weight currents (hence $m\geq m'$),
or proportional to $e_-$ (where $m-m'=-1$). Clearly, only the
latter gives a nonzero contribution. We end up with\col
\[
\int\str\left({\k\over 2}\big(\left(\P_{-m}J^*_z\right)[\P_{m+1}c,e_-]
\big)\right)\,. \]
As the supertrace gives a nondegenerate metric on the super Lie algebra,
integration over $\P_{-m}J^*_z$ gives a delta function for all components
of $\P_{m+1}c$. Note that due to the delta function
$\delta\left(\PLW J^*_z\right)$ in \vgl{eq:RNindactstep1}, no lowest
weight components are present in $\P_{-m}J^*_z$. This agrees with the fact
that there are no highest weights in the commutator with $e_-$. This
concludes the proof.
\end{proof}

Applying the previous lemma, we see that we can set $A_\zb=\t=c=0$ in
\vgl{eq:RNindactstep1}. Then the matrix $\hat\ct$ reduces to a very
simple form. It was discussed in section \ref{sct:RNSBVq} that the
elements of this matrix can be written in terms of the generators of the
quantum cohomology\col\ $\hat{\ct}[\hat{W}^{(j, \alpha_j)}]$. The
equations \re{eq:RNWhatnorm}, \re{eq:RNWAgendef} and \re{eq:RNJhat} show
that\footnote{Any quantum corrections in $\hat{W}$ arise from the
reordering of nonhighest weight currents, which cannot produce a highest
weight current.}\col
\be
\hat{W}^{(j, \alpha_j)}[\PHW J_z,\t\!=\!0,A^*_\zb\!=\!0,c\!=\!0] =
   \cc^j J^{(jj, \alpha_j)}_z\,,\label{eq:RNWhat0}
\ee
where the renormalisation constant $\cc$ is defined in \vgl{eq:RNccdef}.
We will indicate this result with the slightly misleading
notation $\hat{\ct}[\cc\PHW J_z]$.

We can now perform the change of variables alluded to in the introduction.
Passing from the Haar measure $[\d g\, g^{-1}]$ to the measure $[dJ_z]$,
we pick up a Jacobian\col
\be
[\d g g^{-1}]=[dJ_z]\exp \left(-2\tilde{h} S^-[g]\right)\,.
\label{eq:RNchmeas}
\ee

Combining the eqs.\ \re{eq:RNindactstep1}, \re{eq:RNWhat0} and
\re{eq:RNchmeas}, we obtain the induced action in a path integral
formulation for the \WZW\ in the highest weight gauge\col
\bea
\exp \left(-\Gamma[\mu]\right)\!&=&\!
   \int[dJ_z]\
   \delta\left((1-\PHW )( J_z -\frac \k 2 e_-)\right)\nonu
&&\quad\exp\Bigl(
     -\k_c S^-[g] -{1\over 4\pi xy}\int \str\left(\mu \hat{\ct}[\cc J_z]\right)
\Bigr)\,,\label{eq:RNindactHWG}
\eea
where\col
\be
\k_c=\k+2\tilde{h}\,.\label{eq:RNkcdef}
\ee


We are now in a position to study the effective theory of the
(super)gravity theory. The effective action $W[\check{T}]$ \re{eq:RNpi2}
is, for the particular choice of matter sector, given by\col
\bea
\leqn{\exp \left(-W[\check{T}]\right)=\int
[dJ_z][d\mu ]\ \delta\left((1-\PHW )( J_z -\frac \k 2 e_-)\right) }
&&\exp\Bigl( -\k_c S^-[g] -\frac{1}{4\pi xy} \int \str\left(
    \mu(\hat{\ct}[\cc J_z]-\check{T})\right)\Bigr)\,,
\label{eq:RNWeff}
\eea
where $\check{T}\in\KER{+}$. The integration over $\mu$ gives an
additional delta function\col
\be
\delta\left(\hat{\ct}[\cc J_z]-\check{T}\right)\,.
\label{eq:RNWeffconstr}
\ee
This means that we find for the effective action $W[\check{T}]$ simply $\k_c
S^-[g]$ where the groupelements $g$ are determined by the constraints
imposed via the delta-functions.

To make this more explicit, we have to specify which set of generators
$\ct$ we choose for the \WA. As discussed in subsection \ref{ssct:RNWA},
it is convenient to choose $T^{\rm EM}$, defined in \re{eq:RNTEMdef}, for
the Virasoro operator at $(j, \alpha_j)=(1,0)$. We take all other
generators $\hat{\ct}^{(j, \alpha_j)}$ equal to the generators of the
cohomology $\hat{W}^{(j, \alpha_j)}$. We can write this choice of
generators as\footnote{In fact, we take $\hat{\ct}^{(1,0)} =
{1\over 2y}T^{\rm EM}$ to avoid notational difficulties. We will
show in intermezzo \ref{int:RNextranorm} that normalisation factors
of the generators which do not depend on $\k$, do not influence
the coupling constant or wave function renormalisation.}\col
\bea
\hat{\ct}[\cc J_z] &=&
  \exp\left(\sqrt{\cc} e_0\right) \PHW  J_z
  \exp\left(-\sqrt{\cc} e_0\right)+\nonu
&&  {\cc\over 2xy\k}
    \str\big((\PHW \P_0J_z)\ (\PHW\P_0J_z)\big) e_+
\,.\label{eq:RNctbasis}
\eea
To express the constrains imposed on the \WZW\ by \vgl{eq:RNWeffconstr}
in terms of the groupelements of the
\WZW, we should take into account that the quantum currents $J_z$ are
renormalised, see eqs.\ \re{eq:WZWJgroupclassical} and
\re{eq:WZWJgroupquantum}\col
\be
J_z = {\alpha_\k\over 2} \del g\,g^{-1}\,.
\ee
Combining this last equation with eqs.\ \re{eq:RNWeff},
\re{eq:RNWeffconstr}, \re{eq:RNctbasis}, we find the final result\col
\be
W[\check{T}]=\k_c S^-[g]\,,\label{eq:RNBendresult}
\ee
where $\k_c$ is defined in \vgl{eq:RNkcdef}.
The \WZW\ in \vgl{eq:RNBendresult} is constrained by\col
\bea
\leqn{{\alpha_\k\over 2}\del g g^{-1}+
  \frac{\alpha_\k{}^2}{4xy\k}\str \Big(\P_{\rm NA} \left(\del g
g^{-1}\right)\P_{\rm NA}\left(\del g g^{-1}\right)\Big)e_+\ =}
&&{\k\over 2} e_-+\exp\left(
-\ln\sqrt{\cc} e_0\right) \check{T} \exp\left( \ln\sqrt{\cc} e_0\right)\,.
\label{eq:RNconstrWZW1}
\eea
$\P_{\rm NA}\bar{g}$ are those elements of $\P_0 \bar{g}$ which are
highest weight, \ie the centraliser of $\slt$ in $\bar{g}$. Using a
global group transformation\col
\be
g\rightarrow \exp\left( \ln \left(\sqrt{\frac{\alpha_\k}{\k}}\,\right) e_0
\right)g\,,\label{eq:RNrescale}
\ee
which leaves $S^-[g]$ invariant,
we bring the constraints in the standard form used in \cite{bais}\col
\bea
\leqn{\del g g^{-1}+\frac{1}{4xy}
  \str \Big(\P_{\rm NA}\left(\del g g^{-1}\right)\
           \P_{\rm NA}\left(\del g g^{-1}\right)\Big)e_+}
&=&e_-+\sum_{j,\alpha_j}
\frac{2\k^j}{\cc^j\alpha_\k{}^{j+1}}\check{T}^{(j,\alpha_j)}\
{}_{(jj,\alpha_j)}t\,.
\label{eq:RNconstrWZW}
\eea

From \vgl{eq:RNBcpretty}, we get the level $\k$
as a function of the central charge\col
\be
12 y \k=-12 y\tilde{h}-\left(c-\frac 1 2 c_{\rm
crit}\right)- \sqrt{\left(c-\frac 1 2 c_{\rm crit}\right)^2- 24 (d_B-d_F)
\tilde{h}y}\label{eq:RNBvv2}
\ee

We will now reformulate these results in terms of the Legendre transform
$W^{(0)}$ of the classical limit of the induced action, \vgl{eq:RNW0def},
and prove the conjecture \vgl{eq:RNexact}\col
\[
W[\check{T}] = Z_W\, W^{(0)} \left[ Z^{(\check{T})} \check{T} \right]
\,.
\]
It is possible, using the results of \cite{bais}, to find $W^{(0)}$ by
comparing the Ward identities of the constrained \WZW\ to those of the
classical extended Virasoro algebra. We will follow a different
road. Because\col
\be
W_{cl}[\check{T}] = c W^{(0)}[{1\over c}\check{T}]\,,
\label{eq:RNWclW0}
\ee
we can obtain $W^{(0)}$ by taking the large $c$ limit of eqs.\
\re{eq:RNBendresult} and \re{eq:RNconstrWZW}. We begin by observing
that\col
\bea
\frac{\k^j}{\cc^j\alpha_\k{}^{j+1}} \approx  \k^{-1} \approx -{c\over 6y}\,,
&&\mbox{for large $c$}\,,\label{eq:RNlargecvalues}
\eea
where we used \vgl{eq:RNBvv2} in the second step.
Eq.~\re{eq:RNlargecvalues} is valid whatever the values of $\cc$ and
$\alpha_\k$ turn out to be, as their classical limit is fixed to $1$ and
$\k$ respectively. Together with \re{eq:RNBendresult} and
\re{eq:RNconstrWZW}, this gives\footnote{We hope that there can be no
confusion between $\alpha_\k$ and $\alpha_j$.}\col
\be
W[\check{T}] = -{6y (\k+2\tilde{h}) \over c}
W_{cl}[-{c\over 6y} {\k^j\over \cc^j
\alpha_\k{}^{j+1}} \check{T}^{(j,\alpha_j)}\ {}_{(j,\alpha_j)}t]\,.
\ee
Combining this result with \vgl{eq:RNWclW0}, we find\col
\bea
Z_W &=&-6y\left(\k+2\tilde{h}\right)\,,\label{eq:RNZWresult}\\[2mm]
Z^{(\check{T}^{(j,\alpha_j)})}&=&
  -\frac{\k^j}{6y\, \cc^j\alpha_\k{}^{j+1}}\,.\label{eq:RNZTresult}
\eea
Together with \vgl{eq:RNBvv2}, this provides an all-order expression for
renormalisation factors for the chosen normalisation \vgl{eq:RNctbasis}
of the generators of the \WA.
\begin{intermezzo}\label{int:RNextranorm}%
We now discuss the case where a different normalisation from
\vgl{eq:RNctbasis} for the generators of the \WA\ is used. Suppose that
there is an additional factor $\hat{n}^{(j,\alpha_j)}$\col
\[
\hat{\ct}^{(j,\alpha_j)}= \hat{n}^{(j,\alpha_j)}\hat{W}^{(j,\alpha_j)}\,.
\]
This extra factor can depend on $\k$. If we denote its classical limit as
$n^{(j, \alpha_j)}$, we see that the wavefunction renormalisation is
changed to\col
\[
Z^{(\check{T}^{(j,\alpha_j)})}=
  -{n^{(j,\alpha_j)}\over \hat{n}^{(j,\alpha_j)}}
  \frac{\k^j}{6y\, \cc^j\alpha_\k{}^{j+1}}\,.
\]
As discussed in section \ref{sct:RNSBVq}, we expect that all
$\k$--dependence of the normalisation factor is absorbed in $\cc$,
\vgl{eq:RNWhatnorm}, such that the wavefunction renormalisation factor is
not changed with respect to \vgl{eq:RNZTresult}.
\end{intermezzo}

We want to stress here that while the value of
the coupling constant renormalisation is unambiguously determined, the
computation of the value of the wavefunction renormalisation is very
delicate. Indeed, when the gauged WZW model serves as a guideline
\cite{ZFacruud}, we expect that the precise value of the wavefunction
renormalisation depends on the chosen regularisation scheme. As mentioned
before, the computations leading to the quantum effective action are
performed in the operator formalism using point-splitting regularisation.
Within this framework, we should use $\alpha_\k=\k+\tilde{h}$
\re{eq:WZWkzadef}, and $\cc=\k/(\k+\tilde{h})$ \re{eq:RNccdef}. The
wavefunction renormalisation factor \vgl{eq:RNZTresult} simplifies to\col
\be
Z^{(\check{T}^{(j,\alpha_j)})}\ =\
  -{1\over 6y(\k+\tilde{h})}\,.\label{eq:RNZTresultOPE}
\ee
We expect that this result is fully consistent when using the operator
formalism. To provide further support for this claim, we will compare the
results for $\soN$ supergravity of section \ref{sct:RNsoN} to
perturbative computations in section \ref{sct:RNsemiclass}. The
perturbative calculations also rely on operator methods and give results
which are fully consistent with both eqs.\ \re{eq:RNZWresult} and
\re{eq:RNZTresultOPE}. See \cite{S:thesis} for a further discussion of
the wavefunction renormalisation factors.

\section{An example : $\soN$--supergravity\label{sct:RNsoN}}
We will use the supergravity theories based on the
$N$-extended $\soN$ superconformal algebra \cite{NLN=3kniz,NLN=3bersh} as
an example. A realisation of the matter sector, referred to above, is
constructed from gauged $\osp$ \WZW\ in sections \ref{sct:OSPpart1} and
\ref{sct:OSPpart2}.  Applying the results of the previous section,
we can give all--loop results for the effective theory for arbitrary $N$.
Aspects of $N=1$ and $N=2$ supergravity were studied in 
\cite{grixu,sufrac,bo2}. Features of the $N=3$ theory were examined in
\cite{ZFacN=3}, where one--loop results for the effective action were
given.

The induced action $\G [h,\j,A]$ for the $N$-extended $\soN$
supergravity is defined as\col
\bea
\exp \left(-\G [h,\j,A]\right)&=&\Big\langle\exp\biggl(
 -\frac{1}{\pi} \int\left( h T + \j^a G_a + A^{i} U_{i} \right)\biggr)
\Big\rangle\,.\label{eq:RNindn3}
\eea
The OPEs of the $\soN$- superconformal algebra are given in intermezzo
\ref{int:SO(N)superOPEs}.
%We first derive the Ward identities for $\G$.
%The chiral, linearised supergravity transformations with parameters
%$\e,\q^a,\o^i$\col
%\bea
%\d h&=&\bdel \e + \e \del h - \del \e h + 2 \q^a \j_a\,,
%\nonu
%\d \j^a&=&\bdel \q^a + \e \del \j^a - \frac 1 2 \del \e \j^a+\frac 1 2 \q^a\del
%h-\del \q^a h\nonu
%&& +\l_{ab}{}^i(\q^bA_i-\o_i\j^b)\,,
%\nonu
%\d A^i&=&\bdel \o^i+\e \del A^i+\frac b k \l_{ab}{}^i(\del \q^a\j^b-\q^a\del
%\j^b)
%\nonu
%&&-f_{jk}{}^i\o^jA^k-\del\o^ih\,,
%\label{eq:RNtrans1}
%\eea
%are anomalous in the induced theory\col
%\bea
%\d  \G [h,\j,A]&=&-\frac{c}{12\pi}\int\e\del^3h-
%  \frac{b}{2\pi}\int\q^a\del^2\j_a +
%  \frac{k}{2\pi}\int\o^i\del A_i\nonu
%&&-\frac{\g}{\pi} \int \q^a
%    \j^b\P_{ab}^{ij}\left(U_iU_j\right)_{\mbox{\small eff}}\,.
%\label{eq:RNano1} \eea
%The last term is due to the nonlinear nature of the superconformal
%algebras. Defining\col
%\be
%u^i=-\frac{2\pi}{k}\frac{\d\G [h,\j,A ]}{\d A_i}\,
%\ee
%one finds\col
%\bea
%\left(U^{(i}U^{j)}\right)_{\mbox{\small eff}}(x)&=&
%1/\exp \left(-\G\right)\nonu
%&&\Big\langle\int d^2\,x\,U^{(i} U^{j)} (x)\exp\biggl( -\frac{1}{\pi} \int
%\Bigl( h T + \j^a G_a + A^i U_i\Bigr)\biggr)\Big\rangle
%\nonu
%&=&\left(\frac k 2 \right)^2 u^i(x)\,u^j(x)\nonu
%&&+\frac{k\pi}{4}\lim_{y\rightarrow x} \biggl(\frac{\del u^i (x)}{\del
%A_j(y)} -\frac{\del}{\bdel}\d^{(2)}(x-y)\d^{ij} + i\rightleftharpoons j
%\biggr)\,. \label{eq:RNeffreg}
%\eea
%The limit in the last term of \vgl{eq:RNeffreg} reflects the
%point-splitting regularisation of the composite terms in
%\vgl{eq:RNSO(N)algebra}. One notices that in the limit
%$k\rightarrow\infty$, $u$ becomes $k$ independent and one has simply\col
%\be
%\lim_{c\rightarrow\infty}\left(\left( \frac 2 k \right)^2
%\left(U^{(i}U^{j)}\right)_{\mbox{\small eff}}(x)\right)= u^i(x) \, u^j(x)\,.
%\label{eq:RNeffreglc}
%\ee
%Using \vgl{eq:RNeffreg}, we find that \vgl{eq:RNano1} can
%be rewritten as\col \bea
%\d  \G &=& -\frac{c}{12\pi} \int\e\del^3h- \frac{b}{2\pi}\int
%\q^a\del^2\j_a + \frac{k}{2\pi}\int\o^a\del A_a-
%    \frac{k^2\g}{4\pi} \int \q^a \j^b \P_{ab}^{ij} u^i u^j\nonu
%&&-\lim_{y\rightarrow x}\frac{k\g}{2} \int \q^a \j^b \P_{ab}^{ij}
%\biggl(\frac{\del u^i (x)}{\del A^j(y)}
%-\frac{\del}{\bdel}\d^{(2)}(x-y)\d^{ij}\biggr)\,,\label{eq:RNano3}
%\eea
%where the last term disappears for large $k$. The term proportional to
%the integral of $\q^a\j^b\P^{ij}_{ab} u_i u_j$ in \vgl{eq:RNano3} can be
%absorbed by adding a field dependent term in the transformation rule for
%$A$\col
%\be
%\d_{\mbox{\small extra}}A^i=-\frac{k\g}{2}\q^{\,(a}\j^{b)}\P_{ab}^{ij}u_j\,.
%\ee
%Doing this, we find that in the large $k$ limit, the anomaly reduces to
%the minimal one.
%
The Ward identities can be obtained by constructing the chiral
supergravity transformations which give a minimal anomaly for the
transformation of $\Gamma$. Alternatively, one can directly use the OPEs,
\vgl{eq:WIdOPE}. We only list the result. Introducing\col
\bea
t=\frac{12\pi}{c}\frac{\d\G [h,\j,A ]}{\d h}&{\rm and}&
g^a=\frac{2\pi}{\beta}\frac{\d\G [h,\j,A ]}{\d \j_a}\,,
\eea
the Ward identities for $\Gamma$ are\col
\bea
\del^3h&=&\overline{\nabla}t-\frac{3\beta}{c}\left(\j^a\del + 3 \del\j^a\right)g_a
+\frac{6k}{c} \del A^iu_i
\nonu
\del^2 \j^a&=&\overline{\nabla}g^a-\frac{c}{3\beta} \j^at+\l_{ab}{}^i A_ig^b
-\l_{ab}{}^i\left(2\del\j^b+\j^b\del\right)u_i
\nonu
&&-\frac{k^2\g}{2\beta}\P_{ab}^{ij}u_iu_j-
\frac{k\g\pi}{\beta} \P_{ab}^{ij}\lim_{y\rightarrow x}\biggl(\frac{\del u_i
(x)}{\del A^j(y)}
-\frac{\del}{\bdel}\d^{(2)}(x-y)\d_{ij} \biggr)\nonu
\del A^i&=&\overline{\nabla}u^i+
   \frac \beta k \l_{ab}{}^i\j^ag^b+f^{ij}{}_kA^ju^k\,,
\label{eq:RNwi1}
\eea
where\col
\be
\overline{\nabla}\Phi=\left(\bdel-h\del-h_{\Phi}(\del h)\right)\Phi,
\ee
with\col
\be
h_{\Phi}=2,\frac 3 2 ,\ 1 \quad \mbox{ for }\quad \Phi=t,g^a,\ u^a\,.
\ee
The Ward identities provide us with a set of functional differential
equations for the induced action. Because of the explicit dependence of
the Ward identities on $k$, we immediately see that the induced action is
given in a $1/k$ expansion as\col \be
\G [h,\j,A]=\sum_{i\geq 0}k^{1-i} \G^{(i)} [h,\j,A ]\,.\label{eq:RNohwell}
\ee
This definition of $\G^{(i)}$ is different from \vgl{eq:RNcexp}, where an
expansion in the central charge $c$ was used. The relation between these
two parameters is (see intermezzo \ref{int:SO(N)superOPEs})\col
\be
c\ =\ \frac k 2 \frac{6k + N^2-10}{k+N-3}
\label{eq:RNSOnck}
\ee
Inverting this formula would give rise to square roots which we wish to
avoid. Therefore we will work in this section in the large $k$ limit,
which coincides with the large $c$ limit upto some numerical factors.
In this limit, the Ward identities become local and they are solved by
$\G [h, \j, A]=k\G^{(0)} [h, \j, A ]$.

If one defines the Legendre transform of $\G^{(0)} [h,\j,A]$ as\col
\be
W^{(0)}[t,g,u]=\min_{\{h,\j,A\} } \left( \G^{(0)} [h,\j,A ]-
\frac{1}{4\pi}\int \left(h\,t + 4\j^a\,g_a -2  A^i\, u_i
\right)\right)\,,\label{eq:RNlegdef}
\ee
we can view the Ward identities \re{eq:RNwi1} in the large $k$ limit as a
set of functional equations for $W^{(0)}$ \re{eq:RNW0def}.

Consider now the $\osp$ \WZW\ with action $\k S^-[g]$. We will then show
that by imposing the highest weight gauge constraints on its Ward
identities \re{eq:WZWWardId}, we recover the Ward identities of the
$\soN$ supergravity \vgl{eq:RNwi1} in the classical limit \cite{bais}.\\
The Ward identities of a \WZW\ \vgl{eq:WZWWardId} correspond to the zero
curvature condition on the connections $A_{\zb}$ and $u_z$\col
\be
R_{z\zb}=\del A_{\zb} -\bdel u_z-{[}u_z,A_{\zb}{]}=0\,,\label{eq:RNcurv}
\ee
where $u_z$ is defined in \vgl{eq:WZWudef}. We impose the following
constraint on $u_z$\col
\be
u_z \equiv \left( \begin{array}{ccc}
   0 & u_z^{\pp} & u_z^{+b}\\
   1 & 0 & 0 \\
   0 & -u_z^{+a} & u_z^{ab}
\end{array} \right).
\label{eq:RNconstr1}
\ee
Using \vgl{eq:RNconstr1}, we find that some of the components of
$R_{z\zb}=0$ become algebraic equations. Indeed,
$R_{z\zb}^==R_{z\zb}^0=R_{z\zb}^{-a}=0$ for $0\leq a\leq N$ can be solved for
$A_{\zb}^0$, $A_{\zb}^{\pp}$ and $A_{\zb}^{+a}$, giving\col
\bea
A_{\zb}^0&=&\frac 1 2 \del A_{\zb}^=\nonu
A_{\zb}^{\pp}&=&-\frac 1 2 \del^2 A_{\zb}^=+A_{\zb}^=u_z^{\pp}
+ A_{\zb}^{-a}u_z^{+a}\nonu
A_{\zb}^{+a}&=&\del A_{\zb}^{-a}+A_{\zb}^{=}u_z^{+a} - \sqrt{2}\l_{ab}{}^i
A_{\zb}^{-b}u_z^i,
\eea
where\col
\be
u_z^i\equiv\frac{1}{\sqrt{2}}\l_{ab}{}^iu^{ab}_z.
\ee
The remaining curvature conditions $R_{z\zb}^{\pp} = R_{z\zb}^{+a} =
R_{z\zb}^{ab}=0$ reduce to the Ward identities eq. (\ref{eq:RNwi1}) in
the limit $k\rightarrow\infty$ upon identifying\col
\bea
h&\equiv&A_{\zb}^=\nonu
\j^a&\equiv&iA^{-a}_{\zb}\nonu
A^i&\equiv&-\sqrt{2}\left(A_{\zb}^i-A_{\zb}^=u^i_z\right)\nonu
t&\equiv&-2 \left(u_z^{\pp}+u^i_zu^i_z\right)\nonu
g^a&\equiv&iu^{+a}_z\nonu
u^i&\equiv&-\sqrt{2}u^i_z,
\eea
where\col
\be
A_{\zb}^i\equiv\frac{1}{\sqrt{2}}\l_{ab}{}^iA^{ab}_{\zb}.
\ee
This gives for the classical limit of the effective action \vgl{eq:RNW0def}
of the $\soN$--supergravity\col
\bea
\leqn{W^{(0)}\Big[  t= -2(\del g g^{-1})^{\pp} -2(\del g g^{-1})^i (\del
g g^{-1})^i,}
&&g^a= i (\del g g^{-1})^{+a} , u^i= - \sqrt{2}(\del g g^{-1})^{i} \Big]
\ =\ -\frac 1 2  S^-[g]\,,\label{eq:RNrenor2}
\eea
where $S^-[g]$ is the action for the \WZW\ with the constraints given in
\vgl{eq:RNconstr1}.

The effective action $W[t,g,u]$ is defined by\col
\bea
\leqn{\exp \Big(-W[t,g,u]\Big)\ =\ \int [dh][d\j][dA]}
&&\exp\biggl(-\G [h,\j,A ]
+\frac{1}{4\pi}\int\Bigl( h\,t + 4 \j^a\,g_a -2 A^i\, u_i
\Bigr)\biggr)\,.\label{eq:RNqquan}
\eea
One finds to leading order in $k$ (compare with \vgl{eq:RNWclW0})\col
\be
W[t,g,u]=kW^{(0)}[t/k,g/k,u/k]=\k S^-[g]
\ee
and the level $\k$ of the $\asp$ \KA\ is related to the $\soN$
level $k$ as $k\approx- 2\k$ in the large $k$ limit.

We can now perform the analysis given in section \ref{sct:RNallorder} to
find the all--order result for the effective action.
The classical and quantum expressions for the realisation of the $\soN$
\WA\ in terms of the $\asp$ currents was given in sections
\ref{sct:OSPpart1} and \ref{sct:OSPpart2} respectively. In particular, the
normalisation constants can be found in eqs.\ \re{eq:RNBpolys},
\re{eq:RNBTsdef} and \re{eq:RNospnorms}.\\
We do not give the explicit derivation of eqs.\ \re{eq:RNWeff} and
\re{eq:RNWeffconstr} here, as it is completely analogous to the general
method, see \cite{us1} for a detailed calculation. We find that $W[t, g,
u]$ is given by\col
\be
W[t,g,u]=\k_c S^-[g]\,,\label{eq:RNsoNWS}
\ee
where
\be
\k_c=\k+2\tilde{h}=-\frac 1 2 \left( k-7+2N\right)\,,
\ee
where we used $k=-2\k-1$ \re{eq:RNsoNkkappa}.
The WZNW functional in \vgl{eq:RNsoNWS} is constrained by\col
\bea
\left(\del g g^{-1}\right)^=&=&\frac{\k}{\alpha_\k}\nonu
\left(\del g g^{-1}\right)^{-a}&=&\left(\del g g^{-1}\right)^0=0\nonu
\left(\del g g^{-1}\right)^{\pp}+\frac{\alpha_\k}{\k}\left(\del g
g^{-1}\right)^i\left(\del g g^{-1}\right)^i&=&\frac{t}{4\cc \alpha_\k}\nonu
\left(\del g g^{-1}\right)^{+a}&=&\frac{1}{2i \sqrt{\cc}\alpha_\k}g^a\nonu
\left(\del g g^{-1}\right)^{i}&=&\frac{1}{2\sqrt{2}\alpha_\k }u^i\,,
\eea
which is the analogue of \vgl{eq:RNconstrWZW1}. Performing the global
group transformation \vgl{eq:RNrescale}, substituting the value
$\alpha_\k=\k+\tilde{h}$ and comparing with \re{eq:RNrenor2}, we get\col
\be
W[t,g,u]=Z_W W^{(0)}\left[ Z^{(t)}t,Z^{(g)}g,Z^{(u)}u \right],
\ee
where\col
\bea
&&Z_W = -2\k_c= \left( k-7+2N\right)\label{eq:RNsoNZW}\\[2mm]
&&Z^{(t)}=Z^{(g)}=Z^{(u)}=\frac{1}{k+N-3}\,.\label{eq:RNsoNZT}
\eea
This agrees with the results of section \ref{sct:RNallorder} if we
remember that $W^{(0)}$ is in this section defined via a large $k$
expansion. Using $k=-2\k-1$ \re{eq:RNsoNkkappa}, we can convert eqs.\
\re{eq:RNZTresultOPE} and \re{eq:RNZWresult} to this convention by simply
substituting the factor $6y$ by $2$, corresponding to the difference
between the large $c$ and $k$ limit of $\alpha_\k=\k+\tilde{h}$.

\section{Semiclassical evaluation for the linear superconformal algebras
\label{sct:RNsemiclass}}
In the previous section we computed the renormalisation factors for the
nonlinear $\soN$ superconformal algebras by realising them as \WZW s.
On the other hand, we
showed in chapter \ref{chp:FactFree} that for $N=3$ and $4$ these
algebras can be obtained from linear ones by eliminating the dimension
$1/2$ fields and for $N=4$ an additional $U(1)$ factor. Moreover, we
showed in section \ref{sct:FFexamples} that the effective actions $W$ of
the linear theories can be obtained from the linear effective actions
simply by putting the currents corresponding to the free fields to zero,
\vgl{eq:FFWrelation}. In this section we compute the effective actions
for the linear theories in the semiclassical approximation. To facilitate
comparison with the results for the $\soN$ algebras, we first write the
analogues of eqs.\ \re{eq:RNsoNZW} and \re{eq:RNsoNZT} in the large $c$
expansion\col
\bea
&&Z_W = -3\left( k-7+2N\right)\label{eq:RNsoNZWc}\\[2mm]
&&Z^{(t)}=Z^{(g)}=Z^{(u)}=\frac{1}{3(k+N-3)}\,,\label{eq:RNsoNZTc}
\eea
where the relation between $c$ and $k$ is given in intermezzo
\ref{int:SO(N)superOPEs}\col
\be
c\ =\ \frac k 2 \frac{6k + N^2-10}{k+N-3}\,.\label{eq:RNsoNkc}
\ee

Let us first explain the method which we shall use for the semiclassical
evaluation \cite{zamo2,stony}.
We restrict ourselves to linear algebras.
In the semiclassical approximation, the effective action is computed by a
steepest descent method\col
\begin{eqnarray}
\exp\left(-W[\check{T}]\right) &=&
\int [d\mu]\exp\left(-\Gamma [\mu] - \frac{1}{\pi }\int \check{T}\mu\right)\nonu
& \simeq & \exp\left(-\Gamma [\mu_{\rm cl}] - \frac{1}{\pi }\check{T}\mu_{\rm
    cl}\right)
\int[d\tilde{\mu}]
\exp\left( -\frac{1}{2} \tilde \mu \frac{\delta ^2\Gamma [\mu_{\rm cl}]}{\delta
   \mu_{\rm cl}\delta \mu_{\rm cl}}\tilde{\mu}\right)\,,\nonumber\\
&&\label{eq:RNsceffact}
\end{eqnarray}
where $\mu_{\rm cl}[\check{T}]$ is the saddle point value that solves
\begin{equation}
-\frac{\delta \Gamma}{\delta \mu}[\mu_{\rm cl}] = \frac{1}{\pi }\check{T}\,,
\label{eq:RNscu in A}
\end{equation}
and $\tilde \mu$ is the fluctuation around this point.  Therefore, all that
has to be done is to compute a determinant\col
\begin{equation}
W[\check{T}] \simeq W_{\rm cl}[\check{T}] + \frac{1}{2} \log \sdet
  \frac{\delta^2\Gamma [\mu_{\rm cl}]}{\delta \mu_{\rm cl}\delta \mu_{\rm cl}}\,.
  \label{eq:RNscWsemicl}
\end{equation}
To evaluate this determinant, one may use the Ward identities. Schematically,
they have the form\col
\[\overline{D_1}[\mu]\frac{\delta \Gamma }{\delta \mu} \sim \partial _2\mu\]
where on the \lhs\ there is a covariant differential operator, and on the
\rhs\ the term resulting from the anomaly. The symbol $\partial _2$
is standing for a differential operator of possibly higher order (see for
example \vgl{eq:RNwi1} without the nonlinear term).  Taking the
derivative with respect to $\mu$, and transferring some terms to the \rhs\
one obtains\col
\begin{equation}
\overline{D_1}[\mu]\frac{\delta ^2\Gamma }{\delta \mu\delta \mu}
\sim D_2[\check{T}]\,,
\label{eq:RNscDG=D}
\end{equation}
where there now also appears a covariant operator on the \rhs\, with
$\check{T}$ and $\mu$ again related by \vgl{eq:RNscu in A}. The
sought-after determinant is then formally the quotient of the
determinants of the two covariant operators in \vgl{eq:RNscDG=D}.

For the induced and effective actions of fields coupled to affine
currents, the covariant operators are both simply covariant derivatives,
and their determinants are known. Both induce a \WZW\ action. For the 2-D
gravity action $\mu \rightarrow h$ and $\check{T} \rightarrow t$, the operator on
the \rhs\ is $\partial ^3 + t\partial + \partial t$ and the determinant
is given in \cite{zamo2}. A similar computation for the semiclassical
approximation to $W_3$ can be found in \cite{ssvnc}. From these cases,
one may infer the general structure of these determinants. In the gauge
where the fields $\mu$ are fixed, the operator $\overline{D_1}[\mu]$
corresponds to the ghost Lagrangian. In BV language, the relevant piece
of the extended action is $\mu^*\overline{D_1}[\mu]c + b^*\lambda $ and as in
section \ref{sct:quantumHR} the $\mu^*$-field is identified with the
Faddeev-Popov antighost. The determinant is then given by the induced
action resulting from the ghost currents. Since these form the same
algebra as the original currents (at least for a linear algebra), with a
value of the central extension that can be computed, one has\col
\be
\log\sdet\overline{D_1}[\mu] = c_{ghost}\Gamma ^{(0)}[\mu]\,.\label{eq:RNscdetD1}
\ee
The second determinant can similarly be expressed as a functional integral
over some auxiliary $bc$ and/or $\beta \gamma $ system.  Let us, to be
concrete, take $D_2[h] = \frac{1}{4}(\partial ^3 + t\partial + \partial t)$
as an illustration.  Then we have\col
\begin{equation}
(\sdet D_2[t])^{1/2} = \int[d\sigma ] \exp \frac{1}{8\pi }\left(\sigma
(\partial ^3\sigma + t\partial \sigma + \partial (t\sigma ))\right)
\label{eq:RNschLag}
\end{equation}
where it is sufficient to use a single fermionic integral since $D_2$ is
antisymmetric.  We can rewrite this as\col
\[(\sdet D_2[t])^{1/2} = \exp\left({-\tilde W[t]}\right) =
\langle \exp \left(\frac{1}{\pi }\int t H\right) \rangle_\sigma \]
where $H = \frac{1}{4}\sigma \partial \sigma $.

The propagator of the $\sigma $-field fluctuations is given by\col
\be
\sigma (z,\zb)\sigma (w,\bar{w}) = 2 \frac{(z - w)^2}{\bar z - \bar w}
+ \mbox{regular terms} \label{eq:RNscsigmaprop}
\ee
and the induced action has been called $\tilde W$ instead of $\Gamma $ for
reasons that will be clear soon.\newline
The Lagrangian \vgl{eq:RNschLag} has an invariance\col
\bea
\delta t &=& \partial ^3 \omega + 2(\partial \omega )t + \omega \partial
t\nonu
\delta \sigma &=& \omega \partial \sigma - (\partial \omega )\sigma
\eea
and, correspondingly, $\tilde W[t]$ obeys a Ward identity. We find\col
\begin{equation}
\partial ^3\frac{\delta \tilde W[t]}{\delta t} + 2t\partial \frac{\delta
\tilde W[t] }{\delta t} + (\partial t)\frac{\partial \tilde W[t]}{\partial
t} = -\frac{1}{\p}\overline{\partial }t\,.
\label{eq:RNscWIB}
\end{equation}
Another way to obtain this Ward identity \re{eq:RNscWIB} is by evaluating
the OPE of the $H$-operator.  Due to the propagator \vgl{eq:RNscsigmaprop}
this OPE is not holomorphic. One finds\col
\begin{equation}
H(z,\zb)H(0) = -\frac{c'}{4}\left(\frac{z}{\zb}\right)^2 -
\frac{z}{\zb}H(0) - \frac{z^2}{2\zb}\partial H(0) + \cdots\,,
\label{eq:RNscHalgebra}
\end{equation}
where an ellipsis denotes higher order terms in $\zb$. This OPE also
appears in \cite{zamo2}.

Eq. \re{eq:RNscWIB} is nothing but the usual chiral gauge conformal
Ward identity `read backwards', \ie $t \leftrightarrow \frac{\delta
\Gamma [h]}{\delta h} $ and $h \leftrightarrow \frac{\partial \tilde W
[t]}{\partial t}$. We conclude that $\tilde W[t]$ is proportional to the
Legendre transform of $\G^{(0)}$\col
\be
\tilde W[t] = -6c' W^{(0)}[t]\label{eq:RNscWtilde}
\ee
with $c'=2$.

Note that in \cite{zamo2} different bosonic realisations of the algebra
\re{eq:RNscHalgebra} were used. Starting from the action
$\frac{1}{2}\varphi (\partial ^2 + \frac{t}{2})\varphi $ one finds that
$H_\varphi = \frac{1}{4}\varphi ^2$ satisfies \re{eq:RNscHalgebra} with
$c' = -1/2$. This realisation will appear naturally when we discuss
$N=1$. Another realisation, starting from $\varphi _1[\partial ^3 + t\partial +
\partial t]\varphi _2$, has $c' = -4$ and is a bosonic twin of the one we
used. The same algebra also realises a connection with $sl_2$, through
(\cite{zamo2}) $H(z,\zb) = -\frac{z^2}{2} j^+(\zb) + zj^0(\zb) +
\frac{1}{2}j^-(\zb)$\col\ The antiholomorphic components $j^a$ of $H(z, \zb)$
generate an affine $sl_2$ algebra.

The upshot is that whereas the first determinant is proportional to the
classical induced action $\Gamma$, the second one is proportional to
the classical effective action $W$.  The proportionality constants
are pure numbers independent of the central extension of the original
action.  From these numbers the renormalisation factors for the quantum
effective action in the semiclassical approximation follow\col
\bea
W[\check{T}] &\simeq& cW^{(0)}\left[\frac{\check{T}}{c}\right]
-6c' W^{(0)}\left[\frac{\check{T}}{c}\right] -
\frac{c_{ghost}}{2}\Gamma ^{(0)}[\mu_{\rm cl}]
\nonu
&\simeq& \left(c -6c' -
\frac{c_{ghost}}{2}\right)W^{(0)}\left[\frac{\check{T}}{c}\right] +
\frac{c_{ghost}}{2}\check{T}\frac{\partial W^{(0)}}{\partial
\check{T}}\left[\frac{\check{T}}{c}\right]\nonu
&\simeq& \left(c -6c' -
\frac{c_{ghost}}{2}\right)W^{(0)}\left[\frac{\check{T}}{c}(1 +
\frac{c_{ghost}}{2c})\right]\,. \label{eq:RNscWusemicl}
\eea
%In the example of affine KM-currents, the relevant numbers are
%$6k' =- \tilde h$ and $k_{ghost} = -2\tilde h$, with the familiar result
%\footnote{Different calculations give different answers for the field
%renormalisation factor. See the discussion at the end of the previous
%subsection, appendix B, and \cite{ZFacruud}}.\puzzle{weg ?}

For $\Ww_2$-gravity, the results of \cite{zamo2} follow.  We now turn to $N
= 1 \cdots 4$ linear supergravities.

\subsection{$N = 1$}
This case has been treated also in \cite{zamo2,ZFacMP}.
The induced action is
\be
\exp\left(-\Gamma [h,\psi ]\right) =
\langle \exp\left(-\frac{1}{\pi }(hT + \psi G)\right)\rangle\,,
\ee
where $T$ and $G$ generate the $N = 1$ superconformal algebra with central
charge $c$.  The Ward identities read\footnote{All functional derivatives
are left derivatives.}, with $\Gamma = c\Gamma ^{(0)}$\col
\begin{eqnarray}
\big(\overline{\partial } - h\partial - 2(\partial h)\big)
\frac{\delta \Gamma^{(0)}}{\delta h} -
\frac{1}{2}\big(\psi \partial - 3(\partial \psi
)\big)\frac{\delta \Gamma ^{(0)} }{\delta \psi } &=& \frac{1}{12\pi }\partial
^3 h \nonu
\big(\overline{\partial } - h\partial - \frac{3}{2}(\partial h)\big)\frac{\delta
\Gamma ^{(0)} }{\delta \psi } - \frac{1}{2}\psi \frac{\delta \Gamma
^{(0)}}{\delta h} &=& \frac{1}{3\pi } \partial ^2\psi\,.
\label{eq:RNscWI1}
\end{eqnarray}
From this we read off $\overline{D_1}$ and $D_2$ of \vgl{eq:RNscDG=D}\col
\bea
\overline{D_1} &=& \left(\begin{array}{cc}
\overline{\partial} - h\partial - 2(\partial h) & -\frac{1}{2}\psi \partial
+ \frac{3}{2}(\partial \psi )\\
-\frac{1}{2}\psi  & \overline{\partial } - h\partial - \frac{3}{2}(\partial
h)\end{array}\right)\nonu
D_2 &=& \frac{1}{3\pi }\left(\begin{array}{cc}
\frac{1}{4}(\partial ^3 + (\partial \hat t) + 2\hat t\partial ) &
-\frac{1}{2} (\partial \hat y) - \frac{3}{2}\hat g \partial \\
(\partial \hat g) + \frac{3}{2}\hat g\partial  & \partial ^2 + \hat
t/2\end{array}\right)
\eea

We abbreviated $\hat t = -12\pi \frac{\partial \Gamma ^{(0)}[h]}{\partial h} =
t/c$ and $\hat g = g/c$.  For ease of notation, we will drop the hats in
the computation of $\sdet D_2$.

$\overline{D_1}$ gives rise to the ghost-realisation for $N = 1$, so we
have\col
\be
\sdet \overline{D_1} = 15\,\Gamma ^{(0)}[h,\psi ]\,.
\ee
$D_2$ is a (super)antisymmetric operator, as can be seen by rewriting\col
\bea
\partial ^3 + (\partial t) + 2 t \partial &=& \partial ^3 +
\partial t + t\partial \,,\nonu
\frac{1}{2}(\partial g) + \frac{3}{2} g \partial  &=&
\frac{1}{2}\partial g + g \partial \,,\nonu
(\partial g) + \frac{3}{2} g \partial  &=& \frac{1}{2} g
\partial  + \partial g\,.
\eea
The relevant action is then\col
\be
\frac{1}{\pi } \int\left(\frac{1}{8}\sigma (\partial + \partial t +
t\partial )\sigma  - \frac{1}{2}\sigma (\partial g)\varphi - \frac{3}{2}
\sigma g\partial \varphi + \frac{1}{2}\varphi \left(\partial ^2 +
\frac{t}{2}\right)\varphi \right)\,.
\ee
The determinant is\col
\begin{equation}
(\sdet D_2)^{1/2} =
\langle \exp\left(-\frac{1}{\pi }(tH + g\Psi )\right)\rangle =
\exp\left(-\tilde W(t,g)\right)\,,
\label{eq:RNscdet2}
\end{equation}
where $H = \frac{1}{4}\sigma \partial \sigma + \frac{1}{4}\varphi ^2$ and
$\Psi = -\frac{1}{2}(\partial \sigma )\varphi + \sigma \partial \varphi $
and the average is taken in a free field sense with propagators
\vgl{eq:RNscsigmaprop} and\col
\begin{eqnarray}
\langle \varphi (z,\zb)\varphi (w,\bar w)\rangle &=&
\frac{ z - w}{\bar z - \bar w}\,.
\label{eq:RNscphiprop}
\end{eqnarray}
This leads to the OPEs\col
\begin{eqnarray}
H(z,\zb)H(0) &=& -\frac{c'}{4}\frac{z^2}{\zb^2} +
\frac{z}{\zb}H(0) + \frac{1}{2}\frac{z^2}{\zb}\partial
H(0) + \cdots\nonu
H(z,\zb)\Psi (0) &=& \frac{1}{2}\frac{z}{\zb}\Psi (0) +
\frac{1}{2}\frac{z^2}{\zb}\partial \Psi (0) + \cdots\nonu
\Psi (z,\zb)\Psi (0) &=& 2c' \frac{z}{\zb^2} -
 \frac{4}{\zb}H(0) - \frac{2z}{\zb}\partial H(0) + \ldots\,,
\label{eq:RNscHPSIalgebra}
\end{eqnarray}
with the value for the central extension $c' = 2 - \frac{1}{2}$.\newline
The resulting Ward identities for $\tilde W$, \vgl{eq:RNscdet2}, are\col
\bea
(\partial ^3 + (\partial t) + 2t\partial )\frac{\partial \tilde
W}{\partial t} -(2(\del g)+6g\del) \frac{\partial \tilde W}{\partial g} &=&
-\frac{c'}{2\pi }\overline{\partial }t\nonu
(\partial ^2+\frac{t}{2}) \frac{\partial \tilde W}{\partial g}
 +(\,\,(\del g)+\frac{3}{2}g\del) \frac{\partial \tilde W
}{\partial t}&=& -\frac{2c'}{\pi }\overline{\partial }g\,.
\eea
Comparing with \vgl{eq:RNscWI1}, and reverting to the proper normalisation
of $t$ and $g$, we have\col
\be
\tilde W[\hat t,\hat g] =
   6c'\ W^{(0)}[\hat t,\hat g] = 6c'\ W^{(0)}[t/c,g/c]\,,
\ee
where we used\col
\be
W^{(0)}[\hat t,\hat g] = \min_{\{h,\psi \}}\left(\Gamma ^{(0)}[h,\psi ] -
\frac{1}{12\pi }\int h\hat t - \frac{1}{3\pi }\int \hat g \psi \right)\,.
\ee
Putting everything together in \vgl{eq:RNscWusemicl} we find, for $N = 1$,
$c' = 3/2$\col
\be
W[t,g] \simeq c W^{(0)}\left[\frac{t}{c},\frac{g}{c}\right] -
\frac{15}{2}\Gamma ^{(0)}[h,\psi ] -
9 W^{(0)}\left[\frac{t}{c},\frac{g}{c}\right]\,.
\ee
Writing these results as\col
\[W^{(N)}[\Phi ] \simeq Z_W^{(N)}  W^{(0)}[Z_\Phi ^{(N)}\Phi ]\]
we have\col
\bea
&&Z_W^{(1)} \simeq c - \frac{33}{2}\nonu
&&Z_t^{(1)} = Z_g^{(1)} \simeq \frac{1}{c}(1 + \frac{15}{2c})\,.
\eea
For reference, the corresponding equations for $N = 0$ are ($c' = 2$),
\bea
W[t] &=& c W^{(0)} [t/c] - \frac{26}{2} \Gamma ^{(0)}[h] - 12 W^{(0)}
[t/c]\nonu
Z^{(0)}_W &=& c - 25\nonu
Z^{(0)}_t &=& \frac{1}{c}(1 + \frac{13}{c})\,.
\eea
These values are in complete agreement with 
\cite{zamo2,KPZ,sufrac,ZFacMP} and with our 
eqs.\ \re{eq:RNsoNZTc}, \re{eq:RNsoNZTc} and
\vgl{eq:RNsoNkc}.

Before going to $N=2$, we comment on the technique we used to obtain
eqs.\ \re{eq:RNscHalgebra} and \re{eq:RNscHPSIalgebra}. The easiest way
is to expand the fields $\sigma , \varphi $ in solutions of the free
field equations\col
\bea
\sigma (z,\zb) &=& \sigma ^{(0)} (\zb) + z \sigma ^{(1)} (\zb)
+ \frac{z^2}{2} \sigma ^{(2)} (\zb)  \nonu
\varphi(z,\zb) &=& \varphi^{(0)} (\zb) + z \varphi^{(1)} (\zb)
\label{eq:RNantihol}
\eea
and read off the OPEs for the antiholomorphic coefficients from eqs.\
\re{eq:RNscsigmaprop}, \re{eq:RNscphiprop}. Then all singular terms are
given in \vgl{eq:RNscHPSIalgebra}. An alternative would be to use Wick's
method, with the contractions given by the propagators. The resulting
bilocals then give, upon Taylor-expanding, the same algebra as in
\vgl{eq:RNscHPSIalgebra}, up to terms proportional to equations of motion.
This ambiguity was already present in \cite{zamo2}, see also
\cite{ssvnc}. We have simply used an antiholomorphic mode expansion
like in \vgl{eq:RNantihol} in the following calculation. A disadvantage
is, that in this way one loses control over equation of motion terms.

Let us close the $N=1$ case by noting that, just as the antiholomorphic
modes corresponding to \vgl{eq:RNscHalgebra} generate an $\slt$ \KA,
we get an affine $osp (1|2)$ from the modes of $H$ and $\Psi$ of
\vgl{eq:RNscHPSIalgebra}.

\subsection{$N = 2$}
For $N=2$ the extension of the scheme above has two $\Psi$-fields and a
free fermion $\tau$, with $<\tau(x)\tau(0)> = - \frac{1}{\bar{x}}$. This
last field does not contribute to $H$\col
\bea
H    &=& \frac{1}{4} \sigma \partial \sigma
                  + \frac{1}{4} \sum^2_{a=1} \phi^2_a\nonu
\Psi_a &=& - \frac{1}{2} (\partial \sigma ) \phi_a
                + \sigma  \partial \phi_a - \varepsilon _{ab}\phi_b \tau
                \nonu
A    &=& \varepsilon _{ab} \partial \phi_a \phi_b+\sigma \del \tau\,.
\label{eq:RNN=2}
\eea
Note that $\del A$ is proportional to the equations of motion for $\phi_a$
and $\tau$.
Neglecting terms proportional to equations of motion, we find that
in the algebra of \vgl{eq:RNscHPSIalgebra} the first two equations are
supplemented with\col
\begin{eqnarray}
H(z,\zb)A(0) &=& 0 + \cdots\nonumber\\
\Psi _a(z,\zb)\Psi _b(0) &=& \delta _{ab}\left(\frac{2c'z}{\zb^2} -
\frac{4H(0)}{\zb} - \frac{2z}{\zb}\partial H(0)\right) +
\varepsilon _{ab} \frac{z}{\zb}A(0) + \cdots\nonumber\\
A(z,\zb)\Psi _a(0) &=& \frac{\varepsilon _{ab}}{\zb}\Psi _b(0) +
\cdots\nonumber\\
A(z,\zb)A(0) &=& \frac{c'}{\zb^2} + \cdots\,,
\label{eq:RNscHPSI2 algebra}
\end{eqnarray}
and $c' = 2 - 2\cdot \frac{1}{2} = 1$.  With the central charge of the
ghosts being $c_{ghost} = +6$, we can immediately write down the
$Z$-factors for $N=2$\col
\bea
Z_W^{(2)} = c - 9&&
Z_t^{(2)} = Z_g^{(2)} = Z_a \simeq \frac{1}{c}(1 + 3/c)\,,
\eea
which agrees with eqs.\ \re{eq:RNsoNZWc} and \re{eq:RNsoNZTc} using
\vgl{eq:RNsoNkc}.
The algebra of antiholomorphic coefficients of $H,\Psi_a$ and $A$ is now
$osp(2|2)$.

It should be remarked that for $N = 2$ (and higher) the algebra
\vgl{eq:RNscHPSI2 algebra} does not quite reproduce the Ward identities for
the induced action.  Here also, the equations of motion are involved.  The
difference is in the Ward identity\col
\be
\frac{1}{4}(\partial ^3 + 2t\partial + (\partial t))h -
\frac{1}{2}((\partial g_a) + 3g_a\partial )\psi ^a - (\partial A)\cdot u =
\overline{\partial }t\,.
\ee
The last term, as noted below \vgl{eq:RNN=2}, is proportional to equations of
motion of the free part of the action of the auxiliary system, and is not
recovered from the procedure outlined above. We surmise that, as for $N =
0$ and $1$, these terms do not change the result.

\subsection{$N = 3,4$}
For $N = 3$ and $4$, we refrain from writing out the action and
transformation laws, but the same procedure as before is valid (see
\cite{us1} for $N=3$). The algebra of \vgl{eq:RNscHPSI2 algebra} only
changes in that more $\Psi _a$ and $A$ fields are present. The value of the
central charge $c'^{(N)}$ in that algebra can most simply be obtained from
$H(z)H(0)$, since only $\sigma $ and $\varphi _a$ fields contribute to
it\col
\be
c'^{(N)} = 2 - \frac{N}{2}\,.
\ee
The resulting antiholomorphic coefficients constitute the $osp(N|2)$ \KA:
the dimension $1/2$ field contributes no antiholomorphic modes.

The ghost system central charges vanish for $N = 3,4$.  As a result, for
$N = 3$\col
\bea
Z_W^{(3)} = c - 3,&&
Z_t^{(3)} \simeq \frac{1}{c}\,,
\eea
and for $N = 4$, all $Z$-factors are equal to their classical
values.

Now we compare these results for the renormalisation factors with the
results of eqs.\ \re{eq:RNsoNZWc} and \re{eq:RNsoNZTc} for the {\it
nonlinear} algebras, using the result of section \ref{sct:FFexamples}.
According to section \ref{sct:FFexamples}, the respective effective
actions are equal upon putting the appropriate currents to zero.
This means that the other renormalisation factors are the same for the
linear and the nonlinear theory.

Recall that the linear algebras reduce to the nonlinear ones when
eliminating one spin $1/2$ field for $N=3$, and four spin $1/2$ fields
and one spin $1$ field for $N=4$. In this process, the central charge is
modified\col
\bea
c_{nonlinear}^{(3)} &=& c_{linear}^{(3)} - 1/2\nonu
c_{nonlinear}^{(4)} &=& c_{linear}^{(4)} - 3\,.
\eea
Furthermore, the $so(4)$ superconformal algebra is a special case of the
nonlinear $N=4$ algebra discussed in subsection \ref{ssct:FFN=4}. We find
that they coincide for $k^{so(4)} = \tilde{k}_+ = \tilde{k}_-$, where
$\tilde{k}_\pm$ are the $su(2)$-levels used in subsection \ref{ssct:FFN=4}.
With these substitutions, the agreement is complete, both for the overall
renormalisation factor and for the field renormalisations.

For $N=3$ a similar computation was made \cite{ZFacN=3} directly for the
nonlinear supergravity, using Feynman diagrams to compute
the determinants. In that case the classical approximation is not linear in
$c$, but can be written as a power series. The determinant replacing our
$\sdet \overline{D_1}$ is not directly proportional to the induced
action $\G^{(0)}$ as in \vgl{eq:RNscdetD1}.
In fact, this part vanishes since $c_{ghost}=0$ for $N=3$.
Instead, the determinant contains extra terms. These terms are computed in
\cite{ZFacN=3}. They cancel the non-leading terms of the classical
induced action, at least to the extent they are relevant here
(next-to-leading order).  A similar cancellation was also observed in the
computation of the $\Ww_3$ effective action \cite{ZFacW3ssvnb}.
The non-leading contribution and its cancellation with some of the loop
contributions seems to have been overlooked in \cite{ZFacN=3}. We have
recomputed the renormalisation factors for the nonlinear algebra
with the method of \cite{ZFacN=3}, taking into account the non-leading terms
also. We again find agreement with the results obtained above. Note in
particular that all field renormalisation factors are equal. This
alternative computation of the determinants, using Feynmann diagrams,
implicitely confirms our treatment of equation of motion terms in the
Ward identities.


\section{Discussion}
From \vgl{eq:RNBvv2}, one deduces that for generic values of $\k$,
no renormalisation of the coupling constant beyond one loop occurs if
and only if either $d_B=d_F$ or $\tilde{h}=0$ (or both). We get
$d_B=d_F$ for $su(m\pm 1|m)$, $osp(m|m)$ and $osp(m+1|m)$ and
$\tilde{h}=0$, for $su(m|m)$, $osp(m+2|m)$ and $D(2, 1, \alpha)$. Note
that $P(m)$ and $Q(m)$ have not been considered, since we need an
an invariant metric on $\bar{g}$.\puzzle{ \WZW s } The
non-renormalisation of the couplings is reminiscent of the $N=2$
non-renormalisation theorems \cite{nonrenorGSR} and  \cite{Superspace}
(p.358) for extended supersymmetry. These imply that under suitable
circumstances at most one loop corrections to the coupling constants
are present (the wave function renormalisation may have higher order
contributions). Comparing our list with the tabulation
\cite{susyHRFRS} of super \WA s obtained from a (classical)
reduction of superalgebras, we find that many of them, though not all,
have $N=2$ supersymmetry.\\
We first give an example of a theory where no
renormalisations occur, although there is no $N=2$ subalgebra. There is
an $\slt$ embedding in $osp(3|2)$ which gives the $N=1$ super-$\Ww_2$ algebra
of \cite{josestany:sW2}, which contains four fields (dimensions 5/2, 2, 2,
3/2). Although $osp(3|2)$ is in our list, the $N=1$ super-$\Ww_2$ algebra
does not contain an $N=2$ subalgebra. Still, the dimensions seem to fit in
an $N=2$ multiplet. However, we checked using a \Mathematica\
package for super OPEs in $N=2$ superconformal theory \cite{N2SOPEdefs}
that it is not possible to find an associative algebra in $N=2$
superspace with only a dimension $3/2$ superfield with SOPE closing on
itself and a central extension.\\
As an example of the opposite case
(renormalisation but $N=2$), it seems that all superalgebras based on
the reduction of the unitary superalgebras $su(m|n)$ contain an $N=2$
subalgebra, however our list contains only the series $|m-n|\leq 1$.
Clearly, the structural reason behind the lack of renormalisation beyond
one loop remains to be clarified.

%
%The remarkable property that the values of the central extensions of the
%Virasoro and affine subalgebras are related by $c=3 k +n$ with $n \in \Bbb{Z}$,
%is shared by a number of other nonlinear superconformal and
%quasisuperconformal algebras. These contain only one dimension two field, a
%number of (bosonic and fermionic) dimension 3/2 fields, and an affine
%superalgebra by which we will identify them. There are the $osp(m|2n)$ cases
%with $|m-2n-3|\leq 1$ (with $m=3,4$ and $n=0$ treated in this paper, and
%$m=2,n=0$ the ordinary $N=2$ superalgebra) and the $u(n|m)$ cases with
%$|n-m-2| \leq 1$  from the series in \cite{filipzbig}, and further the
%$osp(n|2m)\oplus sl_2$ algebras  with $|n-2m+3|\leq 1$ from  \cite{fradlin}.
%These same algebras arise also (among others) by quantum Drinfeld-Sokolov
%reduction from the list \ref{chp:renormalisations} where there are no
%corrections to the coupling beyond one loop.
%It would be interesting to investigate whether one can extend
%the analysis of the present paper in this direction.

%\sectionappendix
%\section{Renormalisation factors in \WZW s\label{sct:RNWZWfactors}}
%In this appendix we review the relation for \WZW s between the generating
%functional of connected Greens functions with propagating gauge
%fields, $W$, and its classical limit. For a fully regularised
%calculation to one loop, see \cite{ZFacruud}.\puzzle{nog ergens anders
%refereren?}\\ See section \ref{sct:WZW} for conventions on \WZW s.
%
%Consider a \WZW\ with induced action $\G[A_\zb]$. The generating
%functional $W[u_z]$\footnote{The source $u_z$ is obviously different from
%$u_z$ defined in \vgl{eq:WZWudef}. We hope that this does not cause any
%confusion.} is defined by\col \be
%\exp\left( -W[u_z]\right)=
%\int [dA_{\zb}] \exp\left( -\G[A_{\zb}]+\frac{1}{2\pi x}
%  \int  \; str (u_zA_{\zb}) \right)\,.\label{eq:RNwu}
%\ee
%The Legendre transform of $\G^{(0)}[A_{\zb}]$,
%\be
%W^{(0)}[u_z]=\min_{\{ A_{\zb}\} }\left( \G^{(0)}[A_{\zb}]-\frac{1}{2\pi x}\int
%str \left( u_z\,A_{\zb}\right)\right)
%\ee
%is  explicitly given by\col
%\be
%W^{(0)}[u_z\equiv \del g g^{-1}]=S^-[g].\label{eq:RNqquan1}
%\ee
%It can be argued in several ways that $W(u_z)$ is given by\col
%\be
%W(u_z)=\hat{\k}W^{(0)}[Z_{\k}u_z].
%\ee
%The different computations of the renormalised coupling $\hat \k$
%agree with each other, but the values of the current renormalisation
%factor $Z_{\k}$ differ. First we present two different arguments leading
%to the value used in the text. Then, for completeness, we also sketch
%two other lines of reasoning.
%
%In \vgl{eq:RNwu}, we parametrise the integration variable by $A_{\zb}=
%\bdel g g^{-1}$. For the Jacobian, we use\col
%\be
%[dA_{\zb}]=[dgg^{-1}]\sdet \overline{D}[A_{\zb}\equiv\bdel g g^{-1}]=[dgg^{-1}]
%\exp2\tilde{h}S^+[g]\,.\label{eq:RNyoyo}
%\ee
%and obtain
%\be
%\exp\left(-W[u_z]\right)=\int [dg g^{-1}] \exp\left(
%(\k+2\tilde{h})S^+[g] +\frac{1}{2\p x} \int  \; str (u_z\bdel g g^{-1})
%\right)\,. \label{eq:RNpathint}
%\ee
%Since the currents $\bdel g g^{-1}$ form an antiholomorphic affine
%Lie algebra with level $K=-(\k+2\tilde{h})$,
%$W[u_z]$ is the corresponding induced action, cfr.\ \vgl{eq:RNatwelve}.
%Given the relation \re{eq:WZWJgroupquantum} between $\bdel g
%g^{-1}$ and $J_{\zb}$, we conclude that\col
%\be
%W[u_z]=(\k+2\tilde{h})W^{(0)}
%  \left[ -\frac{u_z}{\alpha_{-\k-2\tilde{h}}} \right]
%\ee
%and we identify\col
%\bea
%\hat{\k}&=&\k+2\tilde{h}\nonu
%Z_\k&=&-\frac{1}{\alpha_{-\k-2\tilde{h}}}.
%\eea
%Using the value for $\alpha_\k$ \vgl{eq:WZWkzadef} appropriate for
%operator calculations, we get\col
%\be
%Z_\k=\frac{1}{ \k+\tilde{h}}\,.\label{eq:RNqquan3}
%\ee
%This is the value used in this chapter.
%
%A different argument rests on the invariance of the Haar measure and the
%Polyakov-Wiegmann formula \re{eq:RNpwfor}. Parametrising $u_z$ as\col
%\be
%u_z\equiv (\k+2\tilde{h})\del h h^{-1}\,,
%\ee
%we find, using \vgl{eq:RNpwfor}, that \vgl{eq:RNpathint} becomes\col
%\be
%\exp\left( -W[u_z]\right)=\exp\left( -(\k+2\tilde{h})S^-[h]\right)
%\int [dg g^{-1}] \exp\left( (\k+2\tilde{h})S^+[h^{-1}g]\right)\,.
%\ee
%Assuming that we use a regulator which leaves the Haar measure invariant, we
%can drop the functional integral and we find that\col
%\be
%Z_\k=\frac{1}{ \k+2\tilde{h}}\,.\label{eq:RNvalue1}
%\ee
%This corresponds to the classical value for $\alpha_K$\col\ $\alpha_K=K$.
%
%It is not difficult to reproduce this value by setting up the
%semiclassical computation differently.
%In fact, if in the perturbative calculation we factorise the
%determinants as follows\col
%\be
%\sdet\left(\frac{ D[u_z/\k]}{\overline{D} [A^{\rm cl}_{\zb}(u_z)]}
%\right)=\frac{\sdet D[u_z/\k]\overline{D} [A^{\rm cl}_{\zb}(u_z)]}
%{\left(\sdet \overline{D} [A^{\rm cl}_{\zb}(u_z)]\right)^2}\,,
%\ee
%and we compute the determinants in the numerator with a vector gauge
%invariant regulator, we find again \vgl{eq:RNvalue1}.
%
%This leads us to yet another method to compute $W[u_z]$\col\
%the Knizhnik-Polyakov-Zamolodchikov approach \cite{KPZ}. The covariant
%induced action -- which can be obtained from the computation of \eg
%$\sdet \left(\overline{D}[A_{\zb}] D[A_z]\right)$ with a gauge invariant
%Pauli-Villars regulator -- is given by \be
%\G [A_{\zb}]+\overline{\G}[A_z]-\frac{\k}{2\pi x}\int
%\str\left(A_{\zb}\,A_z\right)\,,\label{eq:RNcovindaction}
%\ee
%where $A_{\zb}$ transforms as in eq. (\ref{eq:RNgf}) and $ \d A_z =\del
%\h + [\h, A_z]$. Now we want to use (\ref{eq:RNcovindaction}) as a
%quantum action. Using the gauge freedom to fix $A_z\equiv \hat{A_z}$
%yields the gauge fixed, BRST-invariant action\col
%\be
%\G[A_{\zb},\hat{A_z},b,c]=\G [A_{\zb}]+\overline{\G}[\hat{A_z}]- \frac{\k}{2\pi
%x}\int \str\left(\hat{A_z} \,A_{\zb}\right)
%-\frac{\k}{2\pi x}\int bD[\hat{A_z}] c\,.
%\ee
%Assuming that the vectorial gauge symmetry is not anomalous (which may be
%guaranteed by the existence of a nilpotent BRST charge) implies that
%$W[\hat{A}_z]$,
%\be
%\exp\left( -W[\hat{A}_z]\right)=\int[dA_{\zb}][db][dc]
%\exp\left( -\G[A_{\zb},\hat{A}_z,b,c]\right)
%\ee
%does not depend on the gauge, \ie is independent of the value of
%$\hat{A}_z$.
%Performing the integral over the ghost fields using the same
%determinant as in \vgl{eq:RNyoyo}, yields\col
%\be
%\int[dA_{\zb}]\exp\left( -\G[A_{\zb}]+
% \frac{\k}{2\p x}\int \str\left( A_{\zb}\,\hat{A_z}\right)\right)=
%\exp\left( -(\k+2\tilde{h})S^-[g]\right),
%\ee
%where $\hat A_z=\del g g^{(-1)}$. This corresponds to the value $Z_K=1/K$.
%\sectionnormal

\thispagestyle{empty}
\begin{center}
\begin{minipage}{12cm}
\begin{center}
{\Large Katholieke Universiteit Leuven\\}
{\Large Faculteit Wetenschappen\\
      Instituut voor Theoretische Fysica}
\end{center}
\end{minipage}
\vfill
\vspace*{2cm}
\begin{minipage}{12cm}
\begin{center}
{\huge\bf An Algorithmic Approach to\\
Operator Product Expansions, $\Ww$-Algebras and $\Ww$-Strings\\}
\end{center}
\end{minipage}
\vfill
\vspace*{3cm}
\begin{minipage}[t]{12.5cm}
\begin{flushright}
   Promotor: {\bf Dr. W. Troost}
\hfill
       Proefschrift ingediend tot\\
       het behalen van de graad van\\
       Doctor in de Wetenschappen\\
       door {\bf Kris Thielemans}\\ \vspace{1.5mm}
\end{flushright}
\end{minipage}
\vfill
{\Large Leuven, juni 1994.}
\end{center}
%\newpage
\clearpage
%\thispagestyle{empty}
%\vspace*{2cm}
%\begin{flushright}
%\begin{minipage}{9cm}
%\setlength{\parindent}{.7cm}
%\it
%``Dit is geen wetenschap, dit is kunst, waarin de ver\-gelij\-kin\-gen
%uiteenvallen in elementen als akkoorden die zich ontbinden, en waarin
%altijd een symmetrie overheerst die hetzij expliciet, hetzij
%multiplex is, maar altijd van een kristallen sereniteit.''\\
%Pandelume \hfill   \\
%\begin{minipage}{9cm}
%\begin{flushright} \rm Jack Vance\\
%                   De stervende aarde
%\end{flushright}
%\end{minipage}
%\end{minipage}
%\end{flushright}
%\newpage
%
\ifdankwoord
\thispagestyle{empty}
%\vspace*{2cm}
%\begin{center}
%\begin{minipage}{10cm}
%\setlength{\parindent}{.7cm}
\section*{Dankwoord}
\vspace{.1cm}
%\small
Het is eindelijk zover. Ik ben aan het dankwoord toe. En dat is mij
alleen maar gelukt dankzij een heel aantal mensen. Maar in de eerste
plaats toch dankzij Kristel. Voor de objectieve waarnemer
overschrijden haar inspanningen het ongelofelijke. Ik wil hier alleen
maar vermelden dat ze de eerste versie (en ook latere gewrochten)
van deze thesis volledig heeft nagelezen. Daarmee is ze
waarschijnlijk de eerste ingenieur die een doctoraat in de theoretische
fysica uitleest. Er zijn zelfs een aantal bladzijden van haar hand
(na revisie weliswaar). Kristel, ook bedankt voor het geduld, de
aandrang, de piano in de morgenstond en alles wat je voor me doet. Het is
mij volslagen onmogelijk om een objectieve waarnemer te zijn, maar
dat vinden we niet erg.

Het was mij zeker ook niet gelukt zonder Walter. Hij wist mij zodanig
te begeleiden dat ik hem al lang niet meer als een promotor zie. Het
is bovendien gemakkelijk als je weet dat er \'e\'en deur verder
iemand zit die al je vragen kan beantwoorden.

Ik wil natuurlijk ook de andere mensen die ik op het Instituut voor
Theoretische Fysica heb leren kennen, danken. Stany voor de nauwe
samenwerking in de eerste helft van mijn doctoraat. Filip voor de
fysica-discussies van het eerste uur en de babbels erna. Alex,
Dirk, Frank, Ruud, Stefan en Toine voor mij altijd bereidwillig hun
idee\"en uit te leggen. Natuurlijk ook Alex die, alhoewel hij ver weg
is, steeds kortbij genoeg is om te helpen. I also want to thank
Zbigniew for the attention he paid to an unknowing freshman. Jose,
your stay here at Leuven has been a very fruitful experience. Thanks
for everything you learned me since then. I'm sure one year London
together will not be long enough. Christine en Anita hebben het zware
werk dikwijls van mij overgenomen. Al de andere medewerkers (te veel
om op te noemen) zijn er in geslaagd om een goede sfeer op te bouwen,
mijn interesse in iets anders dan stringtheorie levend te houden,
en mijn computerexpertise aanzienlijk te laten uitbreiden.
Ik dank profs.\ F.\ Cerulus en A.\ Verbeure in het bijzonder
voor de geboden faciliteiten.

It has been over the last few years my pleasure to get to know
many other theoretical physicists.  I wish to thank them all for the
very good atmosphere there always seems to be between them. In
particular, I want to thank Peter, Sergey, Klaus, Hong, Xujing, Kaiwan
(we Westerners don't know how to write a Chinese name) and Chris for
the scientific contacts, but especially for the friendship.

Ik zou mijn wetenschappelijke roots verloochenen als ik niet aan een
heleboel mensen in Gent zou denken tijdens het schrijven van een
dankwoord. En Pol, als je nog eens een artikeltje wilt schrijven met
iets \Mathematica -achtigs in, je zegt het maar (badminton is ook
goed).

Ik dank ook graag mijn ouders (alle vier) en broers en zussen voor
de jaren ondersteuning. Ook de mensen van het Bhag-ensemble, 't
Notenboompje en vele anderen horen hier thuis, omdat ze af en toe
geduldig luisteren naar mijn verwarrende monologen over wat die
koordjes nu toch zijn, maar vooral voor het samen muziek maken. Want
wat is een stringtheoreet zonder snaren ?

Het is de gewoonte op het einde van een dankwoord een lijntje te
wijden aan je levensgezel(lin). Voor \'e\'en keer zal ik de traditie
niet breken. Zonder jou was het er niet.

\begin{flushright} Kris \end{flushright}
%\end{minipage}
%\end{center}

\else %dankwoord


\thispagestyle{empty}
\vspace*{2cm}
\begin{center}
\begin{minipage}{10cm}
\setlength{\parindent}{.7cm}
\section*{Abstract}
\vspace{.1cm}

String theory is currently the most promising theory to explain the
spectrum of the elementary particles and their
interactions. One of its most important features is
its large symmetry group, which contains the conformal transformations
in two dimensions as a subgroup. At quantum level, the symmetry group
of a theory gives rise to differential equations between correlation
functions of observables. We show that these Ward-identities are equivalent
to Operator Product Expansions (OPEs), which encode the
short-distance singularities of correlation functions with
symmetry generators. The OPEs allow us to determine algebraically
many properties of the theory under study. We analyse
the calculational rules for OPEs, give an algorithm to compute
OPEs, and discuss an implementation in {\sl Mathematica}.\\
There exist different string theories, based on extensions of the
conformal algebra to so-called $W$-algebras. These algebras
are generically nonlinear. We study their OPEs, with as main results
an efficient algorithm to compute the
$\beta$-coefficients in the OPEs, the first explicit construction
of the $W\!B_2$-algebra, and criteria for the factorisation
of free fields in a $W$-algebra.\\
An important technique to construct realisations of
$W$-algebras is Drinfel'd-Sokolov reduction. The method consists of
imposing certain constraints on the elements of an affine Lie algebra.
We quantise this reduction via gauged WZNW-models. This enables us
in a theory with a gauged $W$-symmetry, to compute exactly the correlation
functions of the effective theory.\\
Finally, we investigate the (critical) $W$-string theories
based on an extension of the conformal algebra with one
extra symmetry generator of dimension $N$.
We clarify how the spectrum of this theory forms a minimal model
of the $W_N$-algebra.
\end{minipage}
\end{center}



\newpage

\section*{Notations}
\vspace*{2cm}
\begin{description}
\item[$\dz$, $\db$] : ${d \over dz}$, ${d \over d\zb}$.
\item[$\d_\e$] : infinitesimal transformation with $\e(x)$ an infinitesimal
parameter.
\item[$:T_1 T_2:$] : normal ordening.
\item[OPA] : Operator Product Algebra, subsection \ref{ssct:OPAs}.
\item[OPE] : Operator Product Expansions, section \ref{sct:OPEs}.
\item[$\FULL{T_1(z)T_2(z_0)}$] : the complete OPE.
\item[$ T_1\SING{(z)T_2}(z_0)$] : singular part of an OPE.
\item[${[}T_1T_2{]}_n$] : coefficient of $(z-z_0)^{-n}$ in the OPE
 $\FULL{T_1(z)T_2(z_0)}$,    \vgl{eq:OPEdef}.
\item[$\opefour{{c\over 2}}{0}{2T}{\partial T}$] : list of the operators
at the singular poles in an OPE. Highest order pole is given first. First
order pole occurs last in the list.
\item[$\{T_1T_2 \}_n(x_0)$] : coefficient of
   ${ (-1)^{n-1}\over(n-1)! }\ \dz^{n-1}\delta^{(2)}(x-x_0)$ in the Poisson
   bracket $\{T_1(x)T_2(x_0)\}_{\rm PB}$, subsection \ref{ssct:Poisson}.
\item[$\mo{A}{m} $] : mode of the operator $A$, section \ref{sct:modeAlgebra}.
\item[$L_{\{n\}}$] : is $L_{n_p} \cdots L_{n_1}$,  $L_{\{-n\}}$ reversed order,
chapter \ref{chp:conformalInvariance}.
\item[$\bar{g}$] : Lie algebra, see appendix \ref{app:super} for additional
conventions.
\item[$\hat{g}$] : Ka\v{c}-Moody algebra.
\item[$e_0, e_+,e_-$] : generators of $\slt$.
\item[$\KER{\pm}$] : kernel of the adjoint of $e_\pm$.
\item[$\P$] : projection operator in $\bar{g}$, appendix \ref{app:super}.
\item[$\PHW$] : projection on $\KER{+}$.
\item[$\underline{c}$, $\overline{c}$] : index limitation to generators of strictly
negative, resp.\ positive grading,  section \ref{sct:cohomology}.
\item[$(a)_n$] : Pochhammer symbol,
$(a)_n ={\Gamma(a+n)\over \Gamma(a)} = \prod_{j=0}^{n-1} (a+j)$ ;
$a\in\Bbb{R}$ and $n\in\Bbb{N}$, appendix \ref{app:Combinatorics}.
\end{description}

\fi %dankwoord
\newpage
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%             \lower0.6ex\hbox{$\sim$}}}
%\def\dalemb#1#2{{\vbox{\hrule height .#2pt
%        \hbox{\vrule width.#2pt height#1pt \kern#1pt
%                \vrule width.#2pt}
%        \hrule height.#2pt}}}
%\def\square{\mathord{\dalemb{5.9}{6}\hbox{\hskip1pt}}}

\def\w#1#2{{$\Ww_{#1,#2}$}}

\def\ft#1#2{{\textstyle{{#1}\over{#2}}}}
\def\ket#1{\big| #1\big\rangle}
\def\bra#1{\big\langle #1\big|}
\def\braket#1#2{\big\langle #1\big| #2\big\rangle}
\def\phys{\big|{\rm phys}\big\rangle}
%\def\Z{ \rlap Z\mkern3mu Z}%\ss weg

\chapter{Critical \WS s\label{chp:W-strings}}
\mychapter
In this chapter some aspects of critical \WS s are studied. In an
introduction we review some basic knowledge about the bosonic string which
serves as an example for the generalisation in section \ref{sct:Wstrings}
to strings based on a nonlinear gauge algebra. We discuss the
BRST-quantisation of \WS s. In section \ref{sct:W2s} the simplest possible
classical \WA s \w2s are discussed. These are generated by a Virasoro
operator and a bosonic primary field with dimension $s$. We construct
realisations for all the classical $\Ww_{2, s}$ algebras, and obtain the
corresponding classical BRST operators. We show by example that a graded
structure can be given to these BRST operators by performing canonical
field redefinitions involving the ghost and the matter fields. We find
that these graded classical BRST operators can be promoted to fully
quantum-nilpotent operators by the addition of $\hbar$-dependent terms.
For $s\leq 7$, the spectra of these theories have been studied in
\cite{TAM:hs,TAM:zhao2}. Section \ref{sct:W2sminmod} explores the
relation of the resulting effective Virasoro string theories to certain
$\Ww$ minimal models. Explicit result are given for $s=4,5,6$. In
particular, we show how the highest weight states of the $\Ww$ minimal
models decompose into Virasoro primaries.

The cohomologies of \WS\ theories, and their connection to minimal models,
are indicative of a kind of hierarchical structure of string theories,
which was first articulated in the case of supersymmetric extensions of
string theories by Berkovits and Vafa \cite{vafa}. We examine the
possibility of fermionic higher-dimension extensions of the hierarchical
structure in section \ref{sct:embeddings}.

The results of section \ref{sct:W2sminmod} are published in
\cite{TAMK:Wsminmod}. Sections \ref{sct:W2s} and \ref{sct:embeddings}
are based on \cite{TAMK:Wsquant}.

\section{The bosonic string}

Before studying critical \WS s, we briefly discuss the bosonic
string, concentrating on the BRST approach to string theory. The reader may
consult the general references \cite{GSW,Kaku} for further details. Our
discussion will be restricted to strings with a worldsheet which is a
Riemann surface of zero genus.

The Polyakov action \cite{PolyakovString} for a bosonic string in
$D$ dimensions is given by\col
\be
S[X^\mu,g^{ij}] = -{1\over 4\pi}\int dx^2\,\sqrt{g(x)}
g^{ij}(x)\partial_iX^\mu(x)\partial_jX^\nu(x)\eta_{\mu\nu}\,,
\label{eq:bosstringaction}
\ee
where $g$ is the absolute value of the determinant of $g_{ij}$ and
$\eta_{\mu\nu}$ is the $D$-dimensional Minkowski metric. $x$ are coordinates
on the two-dimensional worldsheet and the values of the fields $X^\mu$ are
coordinates in a
$D$-dimensional flat space. The metric $g_{ij}$ is regarded as a
fluctuating field, although it is classically not propagating because
there are no derivatives of it in the action. In the case of noncritical
strings, considering $g_{ij}$ is essential for a consistent theory.
 Due to the definition of the
\emt\ $T^{ij}$ \re{eq:defT}, the equations of motion of the worldsheet
metric precisely constrain $T^{ij}$ to zero. One can use these equations of
motion to show that the action \re{eq:bosstringaction} is classically
equivalent to the Nambu-Goto action \cite{Nambu,Goto}, which is
proportional to the surface area of the worldsheet of the string.

The action \re{eq:bosstringaction} clearly has general
(worldsheet) coordinate invariance. It is also invariant under local
Weyl rescalings of the metric if $X^\mu$ is assigned a zero scaling
dimension and spin. In the conformal gauge, these gauge invariances are
fixed by\footnote{After a Wick rotation to the euclidean plane.}\col
\be
g^{ij} = \exp(\phi) \delta^{ij}\,,\label{eq:conformalgauge}
\ee
with $\phi$ an arbitrary field. The gauge fixing procedure is
conveniently carried out in the BRST formalism. We introduce two ghost
pairs $(b, c)$ and $(\bar{b}, \bar{c})$, one for each gauge fixing
condition in \vgl{eq:conformalgauge}. The total action becomes (in the
complex basis on the worldsheet)\col\puzzle{normalizatie csts}
\be
S_{gf}[X^\mu,\phi,b,c,\bar{b},\bar{c}]= S[X^\mu, \exp(\phi) \delta^{ij}]
  +{1\over\pi}\int c\db b+\bar{c}\dz \bar{b}\,.
\label{eq:gfbosstringaction}
\ee
The field $\phi$ formally drops out of the action. However, on the
quantum level this is only true when the gauge symmetries, and in
particular the Weyl symmetry, survives at the quantum level. Independence
of the quantum theory on the metric implies that the total \emt\ $T_{\rm
tot}^{ij}$ has a zero expectation value. The holomorphic component $T_{\rm
tot}^{ij}$ is given by\col \be
T_{\rm tot} = T_{\rm mat} + T_{bc}\,,
\ee
where $T_{\rm mat}=\ft12 \dz X^\mu \dz X_\mu$ is the \emt\ of the $X^\mu$
and\col \be
T_{bc} = c\dz b + 2(\dz c)b\,,\label{eq:TghVir}
\ee
which gives $b$ conformal dimension $2$ and $c$ dimension $-1$. $T_{\rm
tot}$ has a central charge $c_{\rm mat} - 26$, see section
\ref{sct:FreeFields}. Exactly the same is true for the antiholomorphic
part $\bar{T}_{\rm tot}$, and we will drop the antiholomorphic symmetry
generators in the rest of this chapter. As each $X^\mu$ contributes $1$
to the central charge, we see that it is only in $26$ dimensions that the
metric decouples from the theory. When this condition is satisfied, the
string theory is said to be ``critical", otherwise the theory suffers
from an anomaly, which can be canceled by introducing an action for the
Liouville mode $\phi$ \cite{PolyakovString}\puzzle{ref Polyakov}. We will
not study non-critical string theory here.

The gauge fixed action \re{eq:gfbosstringaction} is a sum of free field
actions, and it is supplemented by a BRST operator of which we give only
the holomorphic part\col \be
Q = \oint c(T_{\rm mat} +{1\over 2}T_{bc})\,,\label{eq:QVirasoro}
\ee
$Q$ is only nilpotent when the total central charge $c_{\rm mat} - 26$
vanishes. Requiring BRST invariance of the physical states implements the
classical constraint $T=0$ which arose from the equations of motion of the
metric. One then identifies physical states with the elements
of the cohomology of $Q$ in the complex generated by $\{\dz X^\mu, \exp
(k_\mu X^\mu), b,c\}$, where the vertex operators were defined in subsection
\ref{ssct:FreeScalar}. As an example, using the OPEs of section
\ref{sct:FreeFields}, we can compute\col \be
Q(b) = T_{\rm tot}\,.
\ee
This means that $T_{\rm tot}$ is BRST-trivial. Because the operators in
an OPE between two BRST-trivial operators are trivial themselves, we
again find that $T_{\rm tot}$ should have no central extension.

The cohomology of the bosonic string can be computed as follows. Let
$\cx$ be an operator depending only on $X^\mu$, and $\cg$ on ($b,c$). The
action of Q \re{eq:QVirasoro} on the normal ordered product of $\cx$ and
$\cg$ is given by\col  \be
Q(\cx\cg) = \sum_{n\geq 0} [T\cx]_{n+1}\dz^nc \ \cg + \cx\ Q(\cg)\,.
\ee
Now, one has\col
\bea
Q(c) = -\dz c c&{\rm and}&Q(\dz c c) = 0\,.
\eea
Combining these three equations, we see that the field\col
\be
\cx c\label{eq:Wbosstringphysc}
\ee
is physical when $\cal X$ is a primary field of dimension $1$. Also, for a
primary $\cx$ of dimension $h$,
\be
\cx\ \dz c c\label{eq:Wbosstringphysdcc}
\ee
is BRST invariant. However, unless $h=1$, it is a trivial field as it is
proportional to $Q(\cx c)$. Hence, for a primary field of dimension
$1$,  $\cx \dz c c$ is also a physical state. One can prove that all
elements of the cohomology can be written in one of these forms,
\re{eq:Wbosstringphysc} or \re{eq:Wbosstringphysdcc}\footnote{
Note that any derivative of a physical state $\phi$ is BRST trivial.
Indeed, we have that $Q([b,\phi]_1) = [T_{\rm tot} \phi]_1 = \dz
\phi$.} \cite{string162,string149,string165}. \puzzle{of alleen op het
gepaste ghostnr ?,}  Of course, to study the spectrum, we should construct
the fields of dimension $1$ generated by $\{\dz X^\mu, \exp (k_\mu
X^\mu)\}$. This problem was solved in \cite{DDF} by introducing the
spectrum generating DDF-operators. We only mention the tachyonic state
$\exp (k_\mu X^\mu)$, which has mass squared $-k^2=-2$.

A string scattering amplitude on the sphere of physical fields $\Phi_i$ is
given in a pathintegral formalism by\col
\be
\int [dX][db][dc]\, \exp\left(-S_{gf}\right)  b\,\dz b\,\dz^2b\
\Phi_1(x_1)\Phi_2(x_2)\ldots\,,\label{eq:bosstringscat}
\ee
where we only wrote the holomorphic antighosts explicitly. The insertion of the
antighosts corresponds to the zero-modes of the gauge fixing
determinant.\puzzle{refs} This insertion restricts non-zero correlation
functions to fields $\Phi_i$ where the total ghost number adds up to $3$.
By considering two-point functions, we see that the expression
\re{eq:bosstringscat} vanishes unless $\Phi_1\sim \cx c$ and $\Phi_2\sim
\cy \del c c$ (for some $X^\mu$ dependent fields $\cx, \cy$), or vice
versa. This leads us to identify the states in the different sectors of
the cohomology\col
\be
\cx c \sim \cx \del c c\,.\label{eq:bosstringidentify}
\ee
For $n$-point functions, an integration over the moduli of the worldsheet
with punctures at the $x_i$ is implied in \vgl{eq:bosstringscat}.
\puzzle{refs} We do not pursue this topic here, see \cite{GSW,Kaku}.

\section{\WS s\label{sct:Wstrings}}

The bosonic string can be generalised by considering a classical theory
with local gauge symmetries, generated by traceless symmetric tensors. The
analogues of the Weyl invariance imply that the generators can be split in
holomorphic and antiholomorphic components, forming each a copy of a
classical \WA. The local symmetries are then gauge fixed, leading to the
introduction of ghost fields. The gauge-fixed action has a nilpotent
symmetry generated by the classical BRST operator. To quantise the theory,
one must renormalise the symmetry transformation rules and introduce
counterterms, order by order in $\sqrt \hbar$, such that BRST invariance
of the effective action is preserved at the quantum level. The theory is
quantisable if one can carry out the procedure in all orders of $\sqrt
\hbar$. If such procedure is not possible, the theory then suffers from
an anomaly.

In a bosonic string theory with 26 scalars, there is no need to add
quantum counterterms or to modify the transformation rules. This is
because a central charge $c=-26$ from the ghosts is cancelled by the
contributions of the matter scalars. If there were $d\ne 26$ scalars in
the theory, it would still be anomaly free after adding $\sqrt\hbar$
dependent counterterms and modifications of the transformation rules.
These counterterms have the interpretation of background charges in the
matter \emt, with the criticality condition $c=26$ achieved by choosing
appropriate background charges. Also in this case, the matter \emt\ forms a
quantum Virasoro algebra with $c=26$. Thus one can construct
the quantum BRST operator directly from the quantum Virasoro algebra,
\vgl{eq:QVirasoro}.

For the simplest nonlinear algebra, one begins with a theory with
classical $\Ww_3$ symmetry generated by fields $T, W$ of dimension 2
and dimension 3. The classical OPE of the primary current $W$ is given
by\col
\be
W\times W\ =\ \opetwo{ 2 T^2}{\del(T^2)} \, .
\ee
Despite the nonlinearity, it is straightforward to obtain the classical
BRST operator.   One way to realise the classical algebra is in terms
of a scalar field $\varphi$\footnote{Throughout this chapter we use a
normalisation $\varphi\times\varphi=\opetwo{-1}{0}$ to conform with the
literature.} and an arbitrary \emt\ $T_X$ \cite{romans}\col
\bea
T&=&-\ft12 (\del\varphi)^2 + T_X\nonu
W&=& {i\over \sqrt 2}\Big(\ft13 (\del\varphi)^3 +2\del\varphi\, T_X \Big)\,.
\label{eq:W3realeff}
\eea
Here, $T_X$  can be realised in terms of any system with a traceless
symmetric \emt, \eg for the multiscalar realisation of $\Ww_3$ it is the
sum of a number of \emt s of free scalars (possibly with background
charge). With the realisation \vgl{eq:W3realeff}, the theory can be
quantised by adding counterterms and modifying the transformation rules.
The corresponding quantum BRST operator is the same as the one that was
constructed by Thierry-Mieg \cite{mieg} from an abstract quantum $\Ww_3$
algebra with critical central charge $c=100$. The quantum corrections of
the theory can be interpreted as adding background charges to the
classical currents, leading to a quantum realisation of the quantum
$\Ww_3$ algebra at $c=100$ \cite{Pope:W3quant}. Unlike the Virasoro
algebra, the quantum modification of the classical $\Ww_3$ algebra is not
merely reflected by introducing a central charge. The (quantum) OPE of
the primary current $\Ww$ is given by the $\Ww_3$-algebra, which is, with
explicit insertions of $\hbar$, given by\col
\bea
\leqn{\hbar^{-1} W(z)\, W(w)\ =\ {16\over (22+5c)}
       \Big( {2\Lambda \over (z-w)^2} + {\del\Lambda \over z-w}
      \Big)}
&&+ \hbar \Big( {2T\over (z-w)^4} + {\del T\over (z-w)^3} +
   {\ft3{10} \del^2 T \over (z-w)^2}  +
   {\ft1{15} \del^3 T \over z-w}\Big) +
\hbar^2 {c/3\over (z-w)^6} \, ,
\eea
with\col
\be
\Lambda = (TT)-\ft3{10}\hbar\del^2T\,.
\ee
Interactions for the critical $\Ww_3$--string were studied by analogy to the
bosonic string in \cite{TAM:W3scattering,West:W3scat}. Its spectrum was
finally determined in \cite{TAM:W3cohom}.

     The above considerations can be extended to more complicated \WA s. A
discussion of the classical BRST operators for the $\Ww_N$ algebras, and
the structure of the quantum BRST operators, may be found in
\cite{brs2,Gron:WNBRST1,Gron:WNBRST2}.  Detailed results for the quantum
BRST operator for $\Ww_4$ were obtained in \cite{Klaus:W4BRST,zhu:BRST}.  In
general, the quantum $\Ww_N$ BRST operator can be regarded as the
appropriate quantum renormalisation of the classical operator that arises
in an anomaly-free quantisation of the theory.

     The classical BRST operator is derived from the symmetry algebra and
can thus be written in terms of the symmetry generators, irrespective of
the model on which the $\Ww$-symmetry is realised. This is not necessarily
true after quantisation. An example of this can be found in the
$\Ww_3$--string.  In addition to the standard multi-scalar classical
realisations (\ref{eq:W3realeff}), there are four special classical
realisations associated with four Jordan algebras \cite{romans}.  It has
been shown \cite{romans,nour:W3,Jose:JordanW3} that these realisations
cannot be extended to realisations of the quantum $\Ww_3$ algebra. This
does not preclude the possibility to build quantum-consistent
$\Ww_3$--string theories based on these classical realisations of the
symmetry. In other words, the possibility exists that one could still
find quantum nilpotent BRST operators having the classical BRST operators
built from the Jordan realisations as their classical limits. We
explicitly checked that for the simplest case based on the real Jordan
algebra, making a $5$ scalar realisation of the classical $\Ww_3$, it is
not possible to add order $\sqrt\hbar$ corrections to the classical
$\Ww_3$ BRST operator, such that the resulting quantum BRST operator is
nilpotent \cite{TAMK:Wsquant}. Thus it appears that one cannot
consistently quantise $\Ww_3$--strings based on the classical Jordan
realisations of the $\Ww_3$ algebra. This result was obtained for all
four Jordan realisations by Vandoren {\em et al.} using the
Batalin-Vilkovisky quantisation scheme \cite{toine}.

The observation that the quantisation of a theory with gauge symmetries
depends on the particular model, leads us to study string theories where
no quantum \WA\ corresponding to the classical gauge algebra exists. In
\cite{TAM:hs,TAM:zhao2} quantum BRST operators were constructed for
theories with a symmetry algebra formed by $T$ and a dimension $s$
current. Because the field-content of a \w2s classical \WA\ is the
smallest possible, it seems simpler to study \w2s--strings than
$\Ww_N$--strings. While the quantum $\Ww_N$ \WA\ does exist for
arbitrary values of the central charge, deformable quantum algebras
with the same field-content as \w2s exist only for $s=3,4,6$. We will
study a particular type of realisation of critical \w2s--strings
in this chapter.

For more details about \WS s, the reader can consult the reviews
\cite{Hull:Wstringsreview,Pope:Wstringsreview,West:Wstringsreview,%
WAreviewpbks}.


\section{$\Ww_{2,s}$--strings\label{sct:W2s}}

     In this section, we shall investigate higher-spin string theories based
on the classical symmetry algebra generated by $T$ and a bosonic primary
field $W$ of dimension $s$, where $s$ is an integer. Such a closed,
nonlinear, $\Ww_{2,s}$ algebra exists classically for all $s\ge3$. The
classical OPEs of the generators $T$ and $W$ are given by\footnote{ For
even $s$, a generalisation seems possible by adding $2\alpha T^{s/2-1}W$ to
the second order pole in the OPE $W(z)W(w)$ and $\alpha \del(T^{s/2-1}W)$ to
the first order pole, with $\alpha$ some arbitrary constant. In this form
the algebra was called $\Ww_{s/s-2}$ in \cite{Hull:WN/M}. However, one
can always choose generators $T, \tilde{W}= W-\alpha/s^2 T^{s/2}$ such
that $\alpha$ is zero for $\tilde{W}(z)\tilde{W}(w)$. }\col
\bea
T\times T &= & \opetwo{2\,T}{\del T}\nonu
T\times W &= & \opetwo{s\, W}{\del W}\nonu
W\times W &= & \opetwo{2\, T^{s-1}}{\del T^{s-1}}\,.
\label{eq:classW2s}
\eea
It is straightforward to verify that this algebra satisfies the Jacobi
identity at the classical level.

     In the case of a linear algebra $[T_i,T_j]= f_{ij}{}^k T_k$, one knows
that the BRST charge will have the form $Q=c^i \,T_i + \ft12 f_{ij}{}^k \, c^i
\, c^j \, b_k$. In our case, we may interpret the nonlinearity on the
\rhs\ of the OPE $W(z)W(w)$ as $T$--dependent structure constants,
leading to the expectation that the BRST current should have the form\col
\be
J=c\, (T+ T_{\beta\gamma}+\ft12 T_{bc}) + \gamma\, W - \del\gamma\,
\gamma\, b\, T^{s-2}\, ,
\label{eq:W2sclassBRST}
\ee
where the $(b,c)$ are the antighost and ghost for $T$, and $(\beta,\gamma)$
are the antighost and ghost for $W$.  They are anticommuting, and have
dimensions $(2,-1)$ and $(s, 1-s)$ respectively.  The ghost Virasoro operators
are given by \vgl{eq:TghVir} and\col
\be
T_{\beta\gamma} = -s\, \beta\, \del\gamma - (s-1)\, \del\beta\, \gamma \,.
\label{eq:Tghspins}
\ee
Performing the classical OPE, we find that \re{eq:W2sclassBRST} is indeed
nilpotent (the coefficient $-1$ in the last term in \re{eq:W2sclassBRST}
is determined by the nilpotency requirement).

     In order to construct a string theory based on the classical \w2s
symmetry, we need an explicit realisation for the matter currents.  Such a
realisation may be obtained in terms of a scalar field
$\varphi$ and an arbitrary \emt\ $T_X$, which may itself be realised, for
example, in terms of scalar fields $X^\mu$\col
\bea
T &=& -\ft12 (\del\varphi)^2 + T_X\nonu
W &=& \sum_{n=0}^N g_n(s)\, (\del\varphi)^{s-2n}\, T_X^n\,,
\label{eq:W2srealisation}
\eea
where $N=[s/2]$. The constants $g_n(s)$ are determined by demanding that $W$
satisfies \re{eq:classW2s}, and we find that they are given by\col
\be
g_n(s) = s^{-1} (-2)^{-s/2} 2^{n+1} {s \choose 2n}\ .
\end{equation}
Actually, as we shall discuss later, when $s$ is even there is also a
second solution for the constants $g_n(s)$, which is associated
with a ``trivial'' string theory.

     In order to discuss the quantisation of the classical
$\Ww_{2,s}$--string theories, the traditional procedure would be to
undertake an order-by-order computation of the quantum effective action,
introducing counterterms and corrections to the transformation rules in
each order in the loop-counting parameter $\sqrt{\hbar}$, such that BRST
invariance is preserved. Such a procedure is cumbersome and error prone,
but fortunately a more straightforward method is available to us here. We
can simply parametrise all the possible quantum corrections to the BRST
operator, and solve for the coefficients of these terms by demanding
nilpotence at the quantum level using \OPEdefs. Before carrying out
this procedure, we shall first discuss a simplification of the structure
of the BRST operator that can be achieved by performing a canonical
redefinition involving the ghost and the matter fields.

We conjecture that the BRST operator in
\re{eq:W2sclassBRST} can be transformed by canonical field redefinition
into the following graded form\col\puzzle{bewijs a la Bergshoeff ?}
\bea
Q &=& Q_0 + Q_1 \label{eq:W2sgradeBRST}\\
Q_0&=& \oint c(T+T_{\beta\gamma} + \ft12 T_{bc}) \label{eq:W2sQ0}\\
Q_1&=& \oint \gamma\Big (\del \varphi)^s +
  \ft{s^2}2 (\del \varphi)^{s-2}\beta\,\del\gamma\Big) \,.
\label{eq:W2sQ1class}
\eea
Here $Q_0$ has grade $(1, 0)$ and $Q_1$ has grade $(0,1)$, with $(p, q)$
denoting the grading of an operator with ghost number $p$ for the dimension
$2$ ghost system and ghost number $q$ for the dimension $s$ ghost system.
We have $Q_0{}^2=Q_1{}^2=\{Q_0,Q_1\}=0$.

This conjecture is based on the following examples.
For the case of \w23, the field redefinition which acomplishes this was
first described in \cite{TAM:W3scattering}.  At the classical level, the
redefinition is given by\col
\bea
c &\longrightarrow& c -b\, \del\gamma\,\gamma + \sqrt2
i\,\del\varphi\,\gamma \nonumber\\
b &\longrightarrow& b \nonumber\\
\gamma &\longrightarrow& \gamma\nonumber\\
\beta &\longrightarrow& \beta -\del b \,b\,\gamma -
  \sqrt2 i\,\del\varphi\,b \nonumber\\
\varphi &\longrightarrow& \varphi +
   \sqrt2 i\, b\,\gamma \nonumber\\ T_X &\longrightarrow&  T_X \, .
\eea
In the case of $s=4$, we explicitly constructed the field redefinitions
that turn the BRST operator in \re{eq:W2sclassBRST} into the form
\re{eq:W2sgradeBRST},\re{eq:W2sQ0},\re{eq:W2sQ1class}\col
\bea
c \!\!&\longrightarrow&\!\! c - 2 \beta \del\gamma\,\gamma -\ft74 (\del
\varphi)^2 \,\gamma + \ft{21}8 (\del\varphi)^2 b\del\gamma\,\gamma -\ft12
T_X \gamma -\ft54 T_X b\del\gamma\,\gamma \nonumber\\
b \!\!&\longrightarrow&\!\! b \nonumber\\
\gamma \!\!&\longrightarrow&\!\!
    \gamma + 2 b\,\del\gamma\,\gamma \nonumber\\
\beta \!\!&\longrightarrow&\!\!
    \beta + 4 b\,\beta\,\del\gamma + 2 b\,\del\beta\,\gamma +
   \ft74(\del\varphi)^2 \,b + \ft{49}8 (\del\varphi)^2\,\del b \,b\,\gamma
\nonumber\\
&&\!\!+ \ft12 T_X\, b -\ft14 T_X\, \del b\, b \,\gamma + 4 \del b\, b\,
\beta\,\del\gamma\,\gamma + 2 \del b\,\beta\,\gamma \nonumber\\
\varphi \!\!&\longrightarrow&\!\! \varphi - \ft72 \del\varphi\, b\,\gamma \nonumber\\
T_X \!\!&\longrightarrow&\!\! T_X + T_X \,b\,\del\gamma + T_X\,\del b \,\gamma +
\ft12 T_X\,\del b\, b\,\del\gamma\,\gamma + \ft12 \del T_X\, b\,
\gamma \, .\label{eq:W2s4redef}
\eea
The field redefinition becomes more complicated with increasing
$s$. Presumably, this conjecture can be proven along the lines of
\cite{Gron:WNBRST1,Gron:WNBRST2}.

      It is worth mentioning that for $s=2k$ there exists another solution
for the realisation of $W$ given in (\ref{eq:W2srealisation}) in which $W$ can be
written as $\ft1k T^k$.   In this case, there exists a canonical field
redefinition under which the BRST operator in \re{eq:W2sclassBRST}
becomes simply $Q=Q_0$.  It is not surprising that the BRST operator with
this realisation describes the ordinary bosonic string since in this case
the constraint $W=0$ is implied by the constraint $T=0$.  We
shall not consider this case further.

        To quantise the classical \w2s--string and obtain the quantum BRST
operator, we add $\sqrt\hbar$-dependent counterterms to the classical
BRST. In order to do this in a systematic way, it is useful to identify
the $\hbar$ dimensions of the quantum fields. An assignment that is
consistent with the OPEs is\col
\be
\{T_X, \del\varphi, b, c, \beta, \gamma\} \sim
\{\hbar, \sqrt\hbar, \hbar,1,\hbar^{s/2}, \hbar^{1-s/2}\}\,.
\ee
We shall make the assumption that the graded structure of the classical
BRST operator is preserved at the quantum level. For $\Ww_{2,3}$, this
has been explicitly found to be true \cite{TAM:W3scattering}. For $s\ge4$,
there certainly exist quantum BRST operators with the graded structure,
as we shall discuss below. Whether there could exist further quantum BRST
operators that do not posses the grading is an open question.

For the scalar field $\varphi$, the quantum corrections that can be added
to $Q_0$ simply take the form of a background-charge term proportional to
a constant $\alpha$. Its \emt\ becomes\col
\bea
T_\varphi&\equiv&
   -\ft12 (\del\varphi)^2 -\alpha\, \del^2\varphi \label{eq:WTphi}\,.
\eea
Similar modifications to $T_X$ can occur\footnote{For ease of notation, we still
write $T_X$ for the quantum \emt.}. The equation $Q_0{}^2=0$ requires that
the total central charge vanishes\col
\be
0=-26 -2(6s^2-6s+1) + 1+12\alpha^2 + c_X\,,
\ee
with $c_X$ the central charge of $T_X$.
In $Q_1$, the possible quantum corrections amount to\col
\be
Q_1=\oint dz\, \gamma \, F(\varphi,\beta,\gamma)\,,\label{eq:W2sQ1}
\ee
where $F(\varphi, \beta, \gamma)$ is a dimension $s$ operator with ghost
number zero such that its leading-order (\ie classical) terms are given
in (\ref{eq:W2sQ1class}). The precise form of the operator $F(\varphi,
\beta, \gamma)$ is determined by the nilpotency conditions $\{Q_0,
Q_1\}=Q_1{}^2=0$. The quantum BRST operators for $\Ww_{2,s}$ theories
with $s=4$, 5 and 6 were constructed in \cite{TAM:hs}, and the results
were extended to $s=7$ in \cite{TAM:zhao2} and $s=8$ in
\cite{TAMK:Wsquant}. The conclusion of these various investigations is
that there exists at least one quantum BRST operator for each value of
$s$. If $s$ is odd, then there is exactly one BRST operator
of the type discussed. If $s$ is
even, then there are two or more inequivalent quantum BRST operators. One
of these is a natural generalisation to even $s$ of the unique odd-$s$
sequence of BRST operators, see also section \ref{sct:W2sminmod}.

As we discussed earlier, the case $s=3$ corresponds to the
$\Ww_3=\Ww_{DS}\!\!A_2$ algebra, which exists as a closed quantum algebra for
all values of the central charge, including, in particular, the critical
value $c=100$. For $s=4$, it was shown in \cite{TAM:zhao2} that the two
\w24 quantum BRST operators correspond to BRST operators for the
$\Ww_{DS}\!\!B_2$ algebra, which again exists at the quantum level for all
values of the central charge. The reason why there are two inequivalent
BRST operators in this case is that $B_2$ is not simply-laced and so
there are two inequivalent choices for the background charges that give
rise to the same critical value $c=172$ for the central charge
\cite{TAM:zhao2}. Two of the four \w26 BRST operators can similarly be
understood as corresponding to the existence of a closed quantum
$\Ww_{DS}\!G_2$ algebra for all values of the central charge, including
in particular the critical value $c=388$ \cite{TAM:zhao2}. However, the
remaining quantum \w2s BRST operators cannot be associated with any
closed deformable quantum \w2s algebras. For example, the quantum \w25
algebra \cite{bouw:W24} only satisfies the Jacobi identities (up
to null fields) for a discrete set of central-charge values, namely
$c=\{-7,\ft67,-\ft{350}{11}, 134\pm 60\sqrt5 \}$. Since none of these
central charges includes the value $c=268$ needed for criticality, we see
that although the quantum \w25 BRST operator can certainly be viewed as
properly describing the quantised \w25--string, it is not the case that
there is a quantum \w25 symmetry in the \w25--string. This is an explicit
example of the fact that a classical theory can be successfully quantised,
without anomalies, even when a quantum version of the symmetry algebra
does not exist. It appears that the existence of closed quantum \WA s is
inessential for the existence of consistent \WS\ theories.

     As usual, physical fields $\Phi$ are determined by the requirement
that they be annihilated by the BRST operator, and that they be BRST
non-trivial. In other words, $Q\Phi=0$ and $\Phi\ne Q\Psi$ for any
$\Psi$.
There are two different sectors \cite{TAM:hs}. The ``discrete''
physical fields, with zero momentum in the $X^\mu$, will not
be considered here \cite{TAM:hs}. The other sector consists of fields with
continuous $X^\mu$ momentum. Both sectors have only physical fields for
particular values of the $\varphi$-momentum. In this sense, $\varphi$ is
considered a ``frozen'' coordinate, and the $X^\mu$ form coordinates in
the effective spacetime.\\
It was conjectured in \cite{TAM:W3scattering} and \cite{TAM:hs} that all
continuous-momentum physical states for multi-scalar \w2s string theories
can be described by physical operators of the form\col
\be
\Phi_\Delta=c\, U(\varphi, \beta, \gamma)\, V_\Delta(X)\, , \label{eq:Wphysop}
\ee
where $\Delta$ denotes the conformal dimension of the operator $V_\Delta(X)$
which creates an effective spacetime physical state which is a highest
weight field with respect to $T_X$. For simplicity, one can always
take the effective-spacetime operator $V_\Delta(X)$ to be tachyonic, since the
discussion of physical states with excitations in the effective spacetime
proceeds identically to that of bosonic string theory. The interesting
new features of the \WS\ theories are associated with excitations in the
$(\varphi, \beta, \gamma)$ fields. Thus we are primarily concerned with
solving for the operators $U(\varphi, \beta, \gamma)$ that are highest
weight under $ T_\varphi +T_{\gamma, \beta}$ with conformal weights
$h=1-\Delta$, and that in addition satisfy $Q_1(U)=0$. Solving these
conditions for $U(\varphi, \beta, \gamma)$, with $V_\Delta(X)$ being highest
weight under $T_X$ with conformal weight $\Delta=1-h$, is equivalent to
solving the physical-state conditions for $\Phi_\Delta$ in
\vgl{eq:Wphysop}.

\section{Minimal models and \w2s--strings\label{sct:W2sminmod}}

     It has been known for some time that there is a close connection
between the spectra of physical states in \WS\ theories, and certain
Virasoro or $\Ww$ minimal models.  This connection first came to light in
the case of the $\Ww_3$--string \cite{das,TAM:lpsx,TAM:scat,West:W3scat},
where it was found that the physical states in a
multi-scalar realisation can be viewed as the states of Virasoro-type
bosonic strings with central charge $c_X=25\ft12$ and intercepts
$\Delta=\{1,\, \ft{15}{16}, \, \ft12\}$. These quantities are dual to the
central charge $c_{mm}=\ft12$ and weights $h=\{0,\, \ft1{16}, \, \ft12\}$
for the $(p, q)=(3,4)$ Virasoro minimal model, the Ising model, in the
sense that $26=c_X+c_{mm}$, and $1=\Delta+h$. In fact, the physical
operators of the multi-scalar \WS\ have the form \vgl{eq:Wphysop} with
$V_\Delta(X)$ a dimension $\Delta$ field. Further support for this
connection was found in \cite{TAM:W3scattering,West:W3scat} by
considering the scattering of physical states. Using certain
identifications similar to \vgl{eq:bosstringidentify}, it was found that
the S-matrix elements of the lowest $\Ww_3$--string states obey selection
rules also found in the Ising model.

     If one were to look at the multi-scalar $\Ww_N$--string, one would
expect that analogously the physical states would be of the form of
effective Virasoro string states for a $c_X=26-\big(1-{6\over N(N+1)}\big)$
theory, tensored with operators $U(\vec\varphi,\vec\beta,\vec\gamma)$ that
are primaries of the $c_{mm}=1-{6\over N(N+1)}$ Virasoro minimal
model, \ie the $(p,q)=(N,N+1)$ unitary model.  Here, $\vec\varphi$
denotes the set of $(N-2)$ special scalars which, together with the $X^\mu$
appearing in $T_X$, provide the multi-scalar realisation of the $\Ww_N$
algebra.  Similarly, $\vec\beta$ and $\vec\gamma$ denote the sets of
$(N-2)$ antighosts and ghosts for the dimension $3, 4, 5,\ldots,N$
currents. The identification with a particular minimal model is in these
cases based solely on the set of conformal dimensions that occur for the
$U$ field in \vgl{eq:Wphysop}. The rapid growth of the complexity
of the $\Ww_N$ algebras with increasing $N$ means that only incomplete
results are available for $N\ge4$, but partial results and general
arguments have provided supporting evidence for the above connection.

     A simpler case to consider is a $\Ww_{2,s}$--string, corresponding to
the quantisation of the classical theories described in the
previous section.  As already mentioned, there is for any $s$ a
``regular'' BRST operator, which has the feature that the
associated minimal model, with \emt\ $T_{mm}=T_\varphi+T_{\beta\gamma}$,
has central charge\col
\be c_{mm}={2(s-2)\over (s+1)}\,.\label{eq:W2sminmodc}
\ee
This is the central charge of the lowest unitary $\Ww_{s-1}$ minimal
model. We will study the case $s=4$, 5 and 6 in further detail in the
subsections which follow. We will make use of the results of
\cite{TAM:hs} for the quantum BRST operators and the lowest physical
states. Our calculations clarify the connection with the
minimal models \cite{TAMK:Wsminmod}.

     When $s$ is even, there are further ``exceptional'' BRST operators in
addition to the regular one described above.  When $s=4$, there is one
exceptional case, for $s=6$ three \cite{TAM:hs} and for $s=8$ four
\cite{TAMK:Wsquant}. The spectra of these theories are studied in
\cite{TAM:zhao2,TAMK:Wsquant}. They share the feature that a negative
weight for $U(\varphi,\beta,\gamma)$ occurs. This implies correspondingly an
intercept value $\Delta>1$ for the effective spacetime Virasoro string, and
hence the existence of some negative-norm physical states. For some of the
exceptional theories the dimensions of the physical states point towards a
correspondence with Virasoro or $\Ww_{DS}\!B_n$ minimal
models, while for others no connection with minimal models has been found
yet. We will not study the theories related to the exceptional BRST
operators here.

We now outline the procedure which will be followed in the study of \w2s
for $s=4,5,6$. The fields in a minimal model of a \WA\ $\Ww$ are given by
the $\Ww$-descendants of some highest weight fields, among which is the unit operator of
the OPA. This implies that the generators of $\Ww$ are contained in the
set of fields, as they are descendants of $\unity$. Hence, if the
physical states \re{eq:Wphysop} are connected with the $\Ww$-minimal
model, there should be physical states such that the $\varphi, \beta,
\gamma$ dependent parts $U^i$ form a realisation of the generators of
$\Ww$. In this respect it is important to note that we are looking for a
realisation of the \WA\ in the BRST cohomology, \ie up to BRST exact
terms.\\
The $U^i$ which generate the \WA\ should have ghost number zero.
This is because $\Ww$ has always non-zero central charge
\re{eq:W2sminmodc} and the generators generally satisfy
$[U^iU^i]_{2h_i}\sim \unity$. Furthermore, they should depend on
$\varphi$ in such a way that they have well-defined OPEs, \ie the
OPE of $\varphi$  with any other physical state should be meromorphic.
This points to zero $\varphi$-momentum states. This
claim is further supported by an analysis of the spectrum which shows
that $\varphi, \beta, \gamma$ dependent parts $U$ of the $\Ww$-highest weight fields and
their descendants have the same $\varphi$-momentum.\\
To summarise, we should look for operators $U^i(\varphi, \beta, \gamma)$
annihilated by $Q_1$. They have ghost number and $\varphi$-momentum zero.
For the \emt\ of $\Ww$ the obvious candidate for $U$ is\col
\be
T_{mm} = T_\varphi + T_{\beta\gamma}\, , \label{eq:WTmm}
\ee
where $T_\varphi$ and $T_{\beta\gamma}$ are given in \re{eq:WTphi} and
\re{eq:Tghspins}.

\subsection{The \w24--string}
Let us consider first the \w24--string. The BRST operator is
then given by
\re{eq:W2sgradeBRST}, \re{eq:W2sQ0}, \re{eq:WTphi}, \re{eq:W2sQ1}, with
$\alpha^2=\ft{243}{20}$ and the operator $F(\varphi, \beta, \gamma)$ given
by\col \bea
\leqn{F(\varphi,\beta,\gamma)=(\del\varphi)^4 + 4\alpha\, \del^2\varphi\,
(\del\varphi)^2 + \ft{41}5 (\del^2\varphi)^2 + \ft{124}{15}
\del^3\varphi\, \del \varphi +\ft{46}{135} \alpha\, \del^4\varphi}
&& +8 (\del\varphi)^2\, \beta \del\gamma -\ft{16}9\alpha\, \del^2\varphi\,
\beta \del\gamma -\ft{32}9 \alpha\, \del\varphi\,
\beta \del^2 \gamma-\ft45 \beta\, \del^3\gamma  \nonu
&& +\ft{16}3 \del^2\beta\, \del\gamma \,.
\eea

In \cite{TAM:hs}, all physical states up to and including level\footnote{
The level $\ell$ of a state is defined with respect to the standard ghost
vacuum $c_1\gamma_1\cdots \gamma_{s-1}\ket{0}$.} $\ell=9$
in $(\varphi, \beta, \gamma)$ excitations were studied for the
\w24--string. It was found that all the continous-momentum physical states
fall into a set of different sectors, characterised by the value $\Delta$
of the effective spacetime intercept, \vgl{eq:Wphysop}. Specifically, for
the \w24 string, $\Delta$ can take values in the set $\Delta=\{ 1,
\ft{14}{15}, \ft35,\ft13,-\ft25,-2\}$. As one goes to higher and higher
levels $\ell$, one just encounters repetitions of these same intercept
values, with more and more complicated operators $U(\varphi, \beta,
\gamma)$. These operators correspondingly have conformal weights $h$ that
are conjugate to $\Delta$, \ie $h=1-\Delta=\{0, \ft1{15}, \ft25, \ft23,
\ft75, 3\}$. For convenience, table \ref{table:W4} reproduces the results
up to level 9, giving the $(\beta, \gamma)$ ghost number $g$ of the
operators $U(\varphi, \beta, \gamma)$, their conformal weights $h$, and
their $\varphi$ momenta $\mu$.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|r|r|}
\hline
&$\quad{\scriptstyle g}\quad$&${\scriptstyle h}\quad$
&$\mu\ ({\scriptstyle {\rm In\ units\ of}\ \alpha/27)\quad}$\\
\hline
$\ell=0$
&${\scriptstyle 3}$
&${\scriptstyle {1\over15}\quad\ 0 }$
&${\scriptscriptstyle -26\quad -24}$\\
$\ell=0$
&${\scriptstyle 3}$
&${\scriptstyle {1\over15}\quad\ 0 }$
&${\scriptscriptstyle -28\quad -30}$\\
$\ell=1$
&${\scriptstyle 2}$
&${\scriptstyle {2\over3}\quad {2\over5} \quad {1\over15}}$
&${\scriptstyle -20\quad -18\quad -16}$\\
$\ell=2$
&${\scriptstyle 2}$
&${\scriptstyle {7\over5}\ \quad {2\over3}}$
&${\scriptstyle -18\quad -14}$\\
$\ell=3$
&${\scriptstyle 1}$
&${\scriptstyle {2\over3}\quad {1\over15}}$
&${\scriptstyle -10\quad -\phantom{1}8}$\\
$\ell=4$
&${\scriptstyle 1}$
&${\scriptstyle {2\over5}}$
&${\scriptstyle  -\phantom{1}6}$\\
$\ell=5$
&${\scriptstyle 1}$
&${\scriptstyle {7\over5}\ \quad {2\over3}}$
&${\scriptstyle -\phantom{1}6\quad -\phantom{1}4}$\\
$\ell=6$
&${\scriptstyle 0}$
&${\scriptstyle 0}$
&${\scriptstyle 0}$\\
$\ell=7$
&${\scriptstyle 0}$
&${\scriptstyle {1\over15}}$
&${\scriptstyle 2}$\\
$\ell=8$
&${\scriptstyle 0}$
&${\scriptstyle {1\over15}}$
&${\scriptstyle 4}$\\
$\ell=9$
&${\scriptstyle 0}$
&${\scriptstyle 3\ \quad 0}$
&${\scriptstyle 0\quad\phantom{-1}6}$\\
\hline
\end{tabular}
\caption{$U(\varphi,\beta,\gamma)$ operators for the \w24 string of
level $\ell$, ghostnumber $g$, dimension $h$ and $\varphi$-momentum $\mu$.
\label{table:W4}} \end{center}
\end{table}

The explicit expressions for the operators $U(\varphi, \beta, \gamma)$
can be quite complicated, and we shall not give them all here. Some
simple examples are as follows. We find $U=\del^2\gamma\, \del\gamma\,
\gamma\, e^{\mu\varphi}$ at level $\ell=0$; $U=\del\gamma\, \gamma\,
e^{\mu\varphi}$ at $\ell=1$; $U=\Big( 10\, \del\varphi\, \del\gamma\,
\gamma -(\mu+2\alpha)\del^2\gamma\, \gamma \Big) e^{\mu\varphi}$ at
$\ell=2$; and $U=\unity$ at $\ell=6$. The values of the momentum $\mu$ are
given in table \ref{table:W4}.

We wish to identify the dimension $3$ generator $W_{mm}$ of the
associated $\Ww_3$ algebra, realised on the $(\varphi, \beta, \gamma)$
system. We observe from the results in \cite{TAM:hs}
that at level $\ell=9$ there is an operator $U(\varphi, \beta, \gamma)$
with conformal weight $3$, ghost number $g=0$, and momentum $\mu=0$.
Clearly this is the required primary dimension $3$ operator. Its detailed
form is\col
\bea
W_{mm}&=&\sqrt{\ft2{13}} \Big(\ft53\, (\del\varphi)^3 +5\alpha
\,\del^2\varphi\, \del\varphi +\ft{25}4\, \del^3\varphi + 20\, \del\varphi\,
\beta\, \del\gamma\nonu
&&\qquad+12 \, \del\varphi\, \del\beta\, \gamma + 12\, \del^2\varphi\,
\beta\, \gamma + 5\alpha \, \del\beta\, \del \gamma + 3\alpha
\, \del^2\beta\, \gamma\Big)\,,\label{eq:W4spin3}
\eea
where we have given it the canonical normalisation in which\col
\be
W_{mm}(z)W_{mm}(w)\sim {c_{mm}/3\over (z-w)^6}+ \hbox{more}\,,
\ee
with the central charge $c_{mm}=\ft45$.  It is now a straightforward matter
to compute the OPEs of the $T_{mm}$ and $W_{mm}$ currents with
\OPEdefs\ and verify that they do indeed generate the $\Ww_3$ algebra at
$c_{mm}=\ft45$. The only noteworthy point in the verification is that at
the second-order pole in the OPE of $W_{mm}$ with $W_{mm}$ there is an
additional dimension $4$ primary, but BRST-trivial current, $\{Q_1,\beta
\}$.

Having found the currents that generate the $\Ww_3$ algebra, we are now
in a position to see how they act on the operators $U(\varphi, \beta,
\gamma)$ occuring in the physical states of the \w24 string. Of course, we already
know that the operators $U(\varphi, \beta, \gamma)$ are primary fields
under $T_{mm}$. Acting with $W_{mm}$, we find that when $h$ takes values
in the set $\{0,\ft1{15}, \ft25, \ft23\}$, the corresponding operators
are highest weight under $W_{mm}$, \ie $\mo{(W_{mm})}{n}U=0$ for $n >0$,
and $\mo{(W_{mm})}{n}U=w\, U$. We find that the weights are as
follows\col
\be
\begin{array}{ll}
T_{mm}:&\{0,\ft1{15},\ft25,\ft23\}\nonu
{243\over\alpha}\sqrt{\ft{13}8} \, W_{mm} :& \{0,\pm1,0,\pm26\}\,.
\end{array}
\ee
\begin{intermezzo}
To be precise, we find that for $h=\ft1{15}$ the
$W_{mm}$ weight is positive for those operators $U(\varphi, \beta,
\gamma)$ that have ${27\over\alpha}\mu= 4\, \hbox{mod}\, 6$, and negative
when $({\alpha\over 27})^{-1}\mu= 2\, \hbox{mod}\, 6$. Similarly, for
operators with $h=\ft23$, we find the $W_{mm}$ weight is positive when
$({\alpha\over 27})^{-1}\mu= 2\, \hbox{mod}\, 6$, and negative when
$({\alpha\over 27})^{-1}\mu= 4\, \hbox{mod}\, 6$. These results accord
with the observation in \cite{TAM:hs} that there are two independent
towers of $h=\ft1{15}$ operators, and two independent towers of $h=\ft23$
operators, with the screening operator $\beta\, \exp({\ft29
\alpha\varphi})$ generating each tower from its lowest-level member.
\end{intermezzo}
Comparing with the results in \cite{fazam:W3freefield}, we see that these
$T_{mm}$ and $W_{mm}$ weights are precisely those for the lowest
$\Ww_3$ minimal model, with $c_{mm}=\ft45$. The remaining operators
$U(\varphi, \beta, \gamma)$ in the physical states of the \w24 string have
$T_{mm}$ weights $h=\ft75$ and 3. We find that these are not highest weight
under the $W_{mm}$ current. In fact, they are $\Ww_3$ descendant fields;
\puzzle{the dots hier verder, wat staat er}
those with $h=\ft75$ can be written as $\mo{(W_{mm})}{-1}+\cdots$ acting on
operators $U(\varphi, \beta, \gamma)$ with $h=\ft25$, and those with $h=3$
can be written as $\mo{(W_{mm})}{-3}+\cdots$ acting on operators
$U(\varphi, \beta, \gamma)$ with $h=0$.

     The conclusion of the above discussion is that the
$U(\varphi,\beta,\gamma)$ operators appearing in the physical states of the
\w24 string are precisely those associated with the $c_{mm}=\ft45$ lowest
$\Ww_3$ minimal model.  Those with $h=\{0,\ft1{15}, \ft25, \ft23 \}$ are
$\Ww_3$ highest weight fields, whilst those with $h=\ft75$ and 3 are $\Ww_3$
descendants.  Viewed as purely Virasoro fields, they are all primaries.  In
fact, what we are seeing is an explicit example of the phenomenon under
which the set of highest weight fields of a $\Ww$ minimal model decomposes into a larger set
of highest weight fields with respect to the Virasoro subalgebra. In this example, since
$c_{mm}$ is less than $1$, the $W_{mm}$ highest weight fields decompose into a finite
number of Virasoro primaries (namely a subset of the primaries of the
$c_{mm}=\ft45$ 3-state Potts model).  In a more generic example, where the
$\Ww$ minimal model has $c_{mm}\ge 1$, the finite number of $\Ww$
highest weight fields will decompose into an infinite number of Virasoro
primaries, with infinitely many of them arising as $W_{mm}$ descendants.
We shall encounter explicit examples of this when we study the \w2s strings
with $s=5$ and $s=6$.

\subsection{The \w25 string}
Let us now turn to the example of the \w25 string. The operator
$F(\varphi, \beta, \gamma)$ appearing in \vgl{eq:W2sQ1} is given by
\cite{TAM:hs}\col
\bea
\leqn{F(\beta,\gamma,\varphi)=(\del\varphi)^5 +5\alpha\, \del^2\varphi\,
(\del\varphi)^3+\ft{305}8 (\del^2\varphi)^2\, \del\varphi
+\ft{115}6 \del^3\varphi\, (\del\varphi)^2}
&& +\ft{10}3 \alpha\, \del^3\varphi\, \del^2\varphi
+\ft{55}{48}\alpha\, \del^4\varphi\, \del\varphi
+\ft{251}{576} \del^5\varphi +\ft{25}2 (\del\varphi)^3\,\beta\,
\del\gamma +\ft{25}4 \alpha\, \del^2\varphi\, \del\varphi \, \beta\,
\del\gamma\nonu
&&+\ft{125}{16} \del^ 3\varphi\, \beta\, \del\gamma +\ft{325}{12}
\del^2\varphi\, \del\beta\, \del\gamma +\ft{375}{16} \del\varphi\,
\del^2\beta\, \del\gamma -\ft{175}{48}\del\varphi\, \beta\,
\del^3\gamma\nonu
&& +\ft53 \alpha\, \del^3\beta\, \del\gamma
-\ft{35}{48}\alpha\, \del\beta\, \del^3\gamma\,,
\eea
with $\alpha^2=\ft{121}6$.
Here, we have $c_X=25$, and the associated minimal model, with
$c_{mm}=1$, is expected to be the lowest $\Ww_4$ minimal model
\cite{TAM:hs}. Following the same strategy as before, we should
begin by looking amongst the operators $U(\varphi, \beta, \gamma)$
associated with the physical states \vgl{eq:Wphysop} with zero $\varphi$
momentum, and zero ghost number, at dimensions $3$ and $4$. These will be
the candidate dimension $3$ and $4$ primary fields of the $\Ww_4$ algebra.
In \cite{TAM:hs}, all physical states of the \w25 string up to level
$\ell=13$ were obtained. In fact one can easily see that the required
physical states associated with the dimension $3$ and $4$ generators will
occur at levels 13 and 14 respectively. We
find the following expressions for the primary dimension $3$ current
$W_{mm}$ and dimension $4$ current $V_{mm}$ of the $\Ww_4$ algebra at
$c_{mm}=1$\col
\bea
 W_{mm}&\!\!=\!\!&\ft12 (\del\varphi)^3 + \ft32\alpha\,
\del^2\varphi \del\varphi +\ft{31}{12} \del^3\varphi +
\ft{15}2 \del\varphi \beta \del\gamma\nonumber\\
&&+5\, \del\varphi \del\beta \gamma + 5\, \del^2\varphi
\beta \gamma + \ft32\alpha\, \del\beta \del \gamma +
\alpha\, \del^2\beta \gamma , \label{eq:W5prim3}\\
V_{mm}&\!\!=\!\!&{-25\over \sqrt{864}}\Big( (\del\varphi)^4 + 4\alpha\,
\del^2\varphi (\del\varphi)^2 + \ft{2317}{150} (\del^2\varphi)^2 +
\ft{277}{25} \del^3\varphi \del\varphi  +\ft{617}{1650}\alpha\,
\del^4\varphi\nonumber\\
&&\  +\ft{108}{5}\del^2\varphi \del\varphi \beta \gamma
 +20\, (\del\varphi)^2 \beta \del\gamma +\ft{292}{25}
(\del\varphi)^2 \del\beta \gamma +\ft{62}{11}\alpha\, \del^2\varphi
\beta \del\gamma \nonumber\\
&&\ + \ft{2104}{275}\alpha\, \del^2\varphi\del\beta \gamma
+ \ft{216}{55}\alpha\, \del\varphi \del^2\beta \gamma +\ft{378}{55}
\alpha \,\del\varphi \del\beta \del\gamma + \ft{108}{55} \alpha\,
\del^3\varphi \beta \gamma \nonumber\\
&&\ -\ft{44}{15} \beta \del^3\gamma
-\ft{132}{25} \del\beta \del^2\gamma +\ft{321}{25} \del^2\beta
\del\gamma + \ft{544}{75} \del^3\beta \gamma
+\ft{44}5 \del\beta \beta \del\gamma \gamma\Big)\,.
\label{eq:W5prim4}
\eea
We have normalised these currents canonically, so that the coefficient
of the highest-order pole in the OPE of a  dimension $s$ operator with
itself is $c_{mm}/s$, where the central charge $c_{mm}=1$ in the present case.

It is now a straightforward matter to check that $T_{mm}$, $W_{mm}$ and
$V_{mm}$ indeed generate the $\Ww_4$ algebra at $c_{mm}=1$. We find complete
agreement with the algebra given in \cite{blumetal,kauwat} again modulo the
appearance of certain additional primary fields that are BRST exact.
Specifically, we find a dimension $5$ null primary field $Q_1(\beta)$ and
its Virasoro descendants in the OPE of $W_{mm}(z)V_{mm}(w)$, and a
dimension $6$ null primary field $Q_1(30\, \del\varphi\, \beta + 11
\sqrt{6}\, \del\beta)$ and its descendants in the OPE $V_{mm}(z)V_{mm}(w)$.

      Having obtained the currents that generate the $\Ww_4$ algebra, we
may now examine the $U(\varphi,\beta,\gamma)$ operators in the physical
states of the \w25 string, in order to compare their weights with those of
the lowest $\Ww_4$ minimal model. Specifically, this model has highest weight fields
with conformal weights $h=\{ 0, \ft1{16}, \ft1{12},
\ft13,\ft9{16},\ft34,1 \}$.  The results presented in \cite{TAM:hs},
extended to level $\ell=14$, are given in table \ref{table:W5}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|r|r|}
\hline
 &$\quad {\scriptstyle g}\quad $&${\scriptstyle h}\qquad\qquad$
&$\mu\ {\scriptstyle ({\rm In\ units\ of}\ \alpha/22)}$\qquad\\
\hline
$\ell=0$
&${\scriptstyle 4}$
&${\scriptstyle {1\over 12}}\quad\phantom{-}{\scriptstyle {1\over 16}}
\quad{\scriptstyle \phantom{1}0}$
&${\scriptstyle -22}\quad
{\scriptstyle -21}\quad {\scriptstyle -20}$\\
$\ell=0$
&${\scriptstyle 4}$
&$\quad\phantom{-}{\scriptstyle {1\over 16}}
\quad{\scriptstyle \phantom{1}0}$
&${\scriptstyle -22}\quad
{\scriptstyle -23}\quad {\scriptstyle -24}$\\
$\ell=1$
&${\scriptstyle 3}$
&${\scriptstyle {3\over 4}}\quad{\scriptstyle \phantom{-}{9\over 16}
\quad {\scriptstyle \phantom{-1}{1\over 3}}
\quad {\scriptstyle {1\over16}}}$
&${\scriptstyle -18}\quad {\scriptstyle -17}\quad {\scriptstyle -16}\quad
{\scriptstyle -15}$\\
$\ell=2$
&${\scriptstyle 3}$
&${\scriptstyle {25\over 16}}\quad \ {\scriptstyle
{4\over 3}}\quad{\scriptstyle \phantom{1}{3\over 4}}$
&${\scriptstyle -17}\quad
{\scriptstyle -16}\quad {\scriptstyle -14}$\\
$\ell=3$
&${\scriptstyle 2}$
&${\scriptstyle {\scriptstyle \phantom{-}1}\, \quad\phantom{-}{9\over16}}
\quad{\scriptstyle {1\over 12}}$
&${\scriptstyle -12}\quad {\scriptstyle -11}\quad {\scriptstyle -10}
$\\
$\ell=4$
&${\scriptstyle 2}$
&${\scriptstyle {25\over16}}\quad {\scriptstyle {9\over16}}$
&${\scriptstyle -11}\quad {\scriptstyle  -\phantom{1}9} $\\
$\ell=5$
&${\scriptstyle 2}$
&${\scriptstyle 3 }\quad{\scriptstyle {25\over12} }\quad
{\scriptstyle {25\over16}} \quad {\scriptstyle \phantom{1}1}$
&${\scriptstyle -12} \quad {\scriptstyle -10} \quad
 {\scriptstyle -\phantom{1}9 \quad {\scriptstyle -\phantom{1}8}} $\\
$\ell=6$
&${\scriptstyle 1}$
&${\scriptstyle {3\over4}} \quad{\scriptstyle {1\over16}}$
&${\scriptstyle -\phantom{1}6} \quad {\scriptstyle -\phantom{1}5} $\\
$\ell=7$
&${\scriptstyle 1}$
&${\scriptstyle \phantom{-}{1\over3}}$
&${\scriptstyle -\phantom{1}4} $\\
$\ell=8$
&${\scriptstyle 1}$
&${\scriptstyle {4\over3}} \quad{\scriptstyle {9\over16}}$
&${\scriptstyle -\phantom{1}4} \quad {\scriptstyle -\phantom{1}3} $\\
$\ell=9$
&${\scriptstyle 1}$
&${\scriptstyle {49\over16}} \quad{\scriptstyle {25\over16}}
\quad {\scriptstyle \phantom{1}{3\over4}}$
&${\scriptstyle -\phantom{1}5}\quad {\scriptstyle -\phantom{1}3} \quad
{\scriptstyle -\phantom{1}2} $\\
$\ell=10$
&${\scriptstyle 0}$
&${\scriptstyle 0}$
&${\scriptstyle 0}
$\\
$\ell=11$
&${\scriptstyle 0}$
&${\scriptstyle {1\over16}}$
&${\scriptstyle 1}$\\
$\ell=12$
&${\scriptstyle 0}$
&${\scriptstyle {1\over12}}$
&${\scriptstyle 2}$\\
$\ell=13$
&${\scriptstyle 0}$
&${\scriptstyle 3} \quad{\scriptstyle {1\over16}}$
&${\scriptstyle 0}\quad {\scriptstyle \phantom{-1}3}$\\
$\ell=14$
&${\scriptstyle 0}$
&${\scriptstyle 4}\quad{\scriptstyle \phantom{-}{49\over 16}
\quad {\scriptstyle \phantom{}{25\over 12}}
\quad {\scriptstyle \phantom{1}0}}$
&${\scriptstyle 0}\quad {\scriptstyle \phantom{-1}1}\quad {\scriptstyle
\phantom{-1}2}\quad
{\scriptstyle\phantom{-1} 4}$\\
\hline
\end{tabular}
\end{center}
\caption{$U(\varphi,\beta,\gamma)$ operators for the \w25 string
\label{table:W5}} \end{table}
One can see from the results in table \ref{table:W5} that indeed all
the conformal weights of the primary fields of the lowest $\Ww_4$
minimal model occur in the \w25 string.  We find that the
corresponding weights under the $\Ww_4$ currents \re{eq:W5prim3} and
\re{eq:W5prim4} are\col
\be \begin{array}{ll}
T_{mm}:&\{0,\ft1{16},\ft1{12},\ft13,\ft9{16},\ft34,1\},\\[2mm]
{352\over\alpha}\, W_{mm}:& \{0,\pm 1, 0,0, \pm11, \pm32, 0\},\\[2mm]
6912\sqrt{6}\, V_{mm}:& \{0,27,-64,128,-405,1728,-6912\}\,.
\end{array} \ee
\begin{intermezzo}
If $h=\ft1{16}$, the $W_{mm}$--weight
is positive when $({\alpha\over 22})^{-1}\mu= 3\
\hbox{mod}\ 4$, and negative when $({\alpha\over 22})^{-1}\mu= 1\
\hbox{mod}\ 4$.  If $h=\ft9{16}$, the $W_{mm}$-weight
is positive when $({\alpha\over 22})^{-1}\mu= 1\ \hbox{mod}\
4$, and negative when $({\alpha\over 22})^{-1}\mu= 3\ \hbox{mod}\ 4$.  If
$h=\ft34$, for which $({\alpha\over22})^{-1}\mu =12 n -18$ or $12n-14$,
with $n$ a non-negative integer (see \cite{TAM:hs}), the $W_{mm}$ weight is
positive when $n$ is odd, and negative when $n$ is even.
\end{intermezzo}
We have checked that these weights agree with those that one finds using
the highest weight vertex-operators of the $\Ww_N$ minimal models in the
``Miura'' realisations discussed in \cite{faly:WAn}\footnote{After
converting from the non-primary basis of Miura currents to the primary
basis that we are using here.}. The remaining $U(\varphi, \beta, \gamma)$
operators obtained here and in \cite{TAM:hs}, with conformal weights $h$
that lie outside the set of weights for the $\Ww_4$ minimal model,
correspond to $W_{mm}$ and $V_{mm}$ descendant states. In other words,
they are secondaries of the $\Ww_4$ minimal model, but they are primaries
with respect to a purely Virasoro $c_{mm}=1$ model. In this more generic
case, with $c_{mm}\ge 1$, the number of primaries in the purely Virasoro
model will be infinite. Thus if we would go on solving the physical state
conditions at higher and higher levels $\ell$, we would find a set of
operators $U(\varphi, \beta, \gamma)$ with conformal weights $h$ that
increased indefinitely. All those lying outside the set $h=\{0, \ft1{16},
\ft1{12}, \ft13,\ft{9}{16}, \ft34,1\}$ would be given by certain integers
added to values lying in the set, corresponding to $W_{mm}$ and $V_{mm}$
descendant fields.

\subsection{The \w26 string}
In \cite{TAM:hs}, it was found that there are four different nilpotent
BRST operators of the form
\re{eq:W2sgradeBRST},\re{eq:W2sQ0},\re{eq:W2sQ1}, corresponding to
different values of $\alpha$, and hence $c_X$. As usual, we shall
be concerned with the case which seems to be associated with a unitary
string theory. This is given by $\alpha^2=\ft{845}{20}$, implying $c_X
=\ft{174}7$ and hence the $(\varphi, \beta, \gamma)$ system describes a
model with $c=\ft87$. We expect this to be the lowest $\Ww_5$ minimal
model. The operator $F(\varphi, \beta, \gamma)$ in this case takes the form
\cite{TAM:hs}\col \bea
\leqn{
F(\beta,\gamma,\varphi)=(\del\varphi)^6+ 6\alpha \del^2\varphi
(\del\varphi)^4 +\ft{765}7 (\del^2\varphi)^2 (\del\varphi)^2+
\ft{256}7 \del^3\varphi (\del\varphi)^3}
&& +\ft{174}{35}\alpha (\del^2\varphi)^3+\ft{528}{35}\alpha \del^3\varphi
\del^2\varphi \del\varphi +\ft{18}7 \alpha \del^4\varphi (\del\varphi)^2
+\ft{1514}{245} (\del^3\varphi)^2\nonu
&&  +\ft{2061}{245} \del^4\varphi \del^2\varphi+\ft{2736}{1225}
\del^5\varphi \del\varphi+\ft{142}{6125}\alpha \del^6\varphi
+18(\del\varphi)^4 \beta\del\gamma + \ft{72}{5}\alpha \del^2\varphi
(\del\varphi)^2 \beta \del\gamma\nonu
&& +\ft{48}5 \alpha (\del\varphi)^3 \del\beta \del\gamma +\ft{216}5
\del^3\varphi \del\varphi \beta\del\gamma +\ft{1494}{35} (\del^2
\varphi)^2 \beta\del\gamma +\ft{5256}{35} \del^2\varphi \del\varphi
\del\beta \del\gamma\nonu
&&+\ft{324}5 (\del\varphi)^2 \del^2\beta\del\gamma-\ft{72}7
(\del\varphi)^2 \beta\del^3\gamma +\ft{204}{175}\alpha \del^4\varphi\beta
\del\gamma +\ft{192}{25}\alpha \del^3\varphi \del\beta
\del\gamma\nonu
&&+\ft{2376}{175}\alpha \del^2\varphi \del^2\beta
\del\gamma-\ft{144}{175}\alpha \del^2\varphi \beta \del^3\gamma
+\ft{1296}{175}\alpha \del\varphi \del^3\beta
\del\gamma-\ft{576}{175}\alpha \del\varphi \del\beta \del^3\gamma\nonu
&&+\ft{1614}{175} \del^4\beta \del\gamma-\ft{216}{35} \del^2\beta
\del^3\gamma +\ft{144}{1225} \beta\del^5\gamma+\ft{144}{35} \del\beta\beta
\del^2\gamma\del\gamma.
\eea

In \cite{TAM:hs}, physical states in the theory up to and including
level $\ell=6$ were studied. Here, we are primarily concerned with
finding the physical states associated with the expected dimension $3,4,5$,
primary fields of the $\Ww_5$ minimal model. These should
occur at levels $\ell=18$, 19 and 20 respectively. It is a straightforward
matter to solve for such physical states \vgl{eq:Wphysop} with
$U(\varphi, \beta, \gamma)$ having zero ghost number and zero $\varphi$
momentum. We find the following results for the dimension $3,4,5$
operators $W_{mm}$, $V_{mm}$ and $Y_{mm}$\col
\beastar
W_{mm}\!\!&=&\!\!\sqrt{\ft2{57}}\Big(\ft73 (\del\varphi)^3 + 7\alpha
\del^2\varphi \del\varphi +\ft{185}{12} \del^3\varphi +
42 \del\varphi \beta \del\gamma\nonumber\\
&&\  +30  \del\varphi \del\beta \gamma + 30  \del^2\varphi
\beta \gamma + 7\alpha \del\beta \del \gamma +
 5\alpha \del^2\beta \gamma\Big) \nonu
V_{mm}\!\!&=&\!\!-\sqrt{\ft7{60819}}\Big(\ft{427}8 (\del\varphi)^4 + \ft{427}2\alpha
\del^2\varphi (\del\varphi)^2 + \ft{10619}{8} (\del^2\varphi)^2 +
743 \del^3\varphi \del\varphi\nonumber\\
&&\  + \ft{3313}{156}\alpha
\del^4\varphi+1455\del^2\varphi \del\varphi \beta \gamma
 +1281 (\del\varphi)^2 \beta \del\gamma +825
(\del\varphi)^2 \del\beta \gamma \nonumber\\
&&\  +\ft{5370}{13}\alpha \del^2\varphi
\beta \del\gamma+ \ft{6900}{13}\alpha \del^2\varphi
\del\beta \gamma
+ \ft{2910}{13}\alpha \del\varphi \del^2\beta \gamma +\ft{4656}{13}
\alpha \del\varphi \del\beta \del\gamma\nonumber\\
&&\  + \ft{1455}{13} \alpha
\del^3\varphi \beta \gamma -247 \beta \del^3\gamma
-494 \del\beta \del^2\gamma +\ft{6891}{7} \del^2\beta
\del\gamma\nonumber\\
&&\  +\ft{7785}{14} \del^3\beta \gamma
+1170 \del\beta \beta \del\gamma \gamma\Big)
\eeastar
\bea
\lefteqn{Y_{mm}=\sqrt{\ft{7}{122}}\Big( \ft{749}{165} (\del\varphi)^5 +
\ft{749}{33}\alpha (\del\varphi)^3 \del^2\varphi + \ft{6091}{22}
(\del^2\varphi)^2 \del\varphi + \ft{1361}{66}\alpha
\del^3\varphi \del^2\varphi}\nonumber\\
&&\quad +\ft{14351}{132} \del^3\varphi (\del\varphi)^2 +
\ft{2330}{429}\alpha \del^4\varphi \del\varphi  +\ft{4825}{1848}
\del^5\varphi + \ft{1498}{11}(\del\varphi)^3 \beta \del\gamma \nonumber\\
&&\quad +\ft{2570}{33} (\del\varphi)^3 \del\beta \gamma  +
\ft{11382}{143}\alpha \del^2\varphi \del\varphi \beta
\del\gamma + \ft{15670}{143}\alpha \del^2\varphi \del\varphi
\del\beta \gamma\nonumber\\
&&\quad + \ft{430}{11}\alpha (\del^2\varphi)^2 \beta
\gamma  +\ft{4270}{143}\alpha \del^3\varphi \del\varphi \eta
\gamma +\ft{2350}{11} \del^2\varphi (\del\varphi)^2 \beta
\gamma \nonumber\\
&&\quad + \ft{7840}{143}\alpha (\del\varphi)^2 \del\beta
\del\gamma +\ft{4380}{143}\alpha (\del\varphi)^2 \del^2\beta
\gamma -52 \del\varphi \beta \del^3\gamma -\ft{624}{7}
\del\varphi \del\beta \del^2\gamma\nonumber\\
&&\quad  + \ft{17331}{77}
\del\varphi \del^2\beta \del\gamma +\ft{9775}{77} \del\varphi
\del^3\beta \gamma -\ft{624}{7} \del^2\varphi \beta \del^2\gamma +
\ft{18541}{77}  \del^2\varphi \del\beta \del\gamma\nonumber\\
&&\quad  + \ft{3390}{11} \del^2\varphi \del^2\beta \gamma+ \ft{7957}{154}
\del^3\varphi \beta \del\gamma+\ft{77705}{462} \del^3\varphi\del\beta
\gamma +\ft{11575}{462} \del^4\varphi \beta \gamma\nonumber\\
&&\quad  -\ft{26}{3}\alpha
\del\beta  \del^3\gamma -\ft{104}{7}\alpha \del^2\beta
\del^2\gamma +\ft{6775}{1001}\alpha \del^3\beta \del\gamma
+\ft{5365}{1001} \alpha \del^4\beta \gamma\nonumber\\
&&\quad  +120 \del\varphi  \del\beta \beta \del\gamma \gamma
+\ft{120}{13}\alpha \del^2\beta \beta \del\gamma \gamma\Big).
\eea
We have as usual given these currents their canonical normalisations. We
have checked that they indeed, together with $T_{mm}=T_\varphi +
T_{\beta\gamma}$, generate the $\Ww_5$ algebra, given in \cite{Klaus:WA5},
with central charge $c_{mm}=\ft87$. Again, one finds additional BRST exact
fields appearing on the right-hand sides of the OPEs of the primary
generators. These null fields are primaries (and their descendants)
except in the case of the OPE $Y_{mm}(z) Y_{mm}(w)$. The new field
occuring at the second order pole of this OPE is only primary up to BRST
exact terms.

     It was found in \cite{TAM:hs} that the physical states of the \w26
string were associated with operators $U(\varphi,\beta,\gamma)$ whose
conformal weights included those of the highest weight fields of the lowest
$\Ww_5$ minimal model, which has $c=\ft87$. Indeed, here we find that the
highest weight fields have the weights \be
 \begin{array}{ll}
T_{mm}:& \{0,\ft2{35},\ft3{35}, \ft27, \ft{17}{35},\ft{23}{35},
\ft45, \ft67,\ft65 \}\\[2mm]
{325\over\alpha} \sqrt{\ft{57}{8}} W_{mm}:& \{0,\pm2, \pm1, 0,
\pm13, \pm39, \pm76, 0, \pm38\} \\[2mm]
25\sqrt{\ft{141911}3}\, V_{mm}:& \{0,11,-14, 50, -74, 66, 836, -1100,
-2299\}\\
{89375\over\alpha} \sqrt{\ft{427}{32} }\, Y_{mm}:& \{ 0,\pm 11, \mp
48, 0, \pm314, \mp 902, \pm 1452, 0, \mp16621\}\,.
\end{array}
\label{eq:W6weights}
\ee
Note that the $\pm$ signs for the weights under the dimension $3$
operator $W_{mm}$ are correlated with those for the weights under the
dimension operator $Y_{mm}$. Again we have checked that these weights
agree with those calculated from the realisations of the $\Ww_N$ minimal
models given in \cite{faly:WAn}. All the physical states of the \w26
string are presumably associated with operators $U(\varphi, \beta,
\gamma)$ that are either highest weight under the $\Ww_5$ algebra, as
given in \vgl{eq:W6weights}, or they are $W_{mm}$, $V_{mm}$ or $Y_{mm}$
descendants of such operators. Some examples of descendant operators can
be found in \cite{TAM:hs}. Again one expects, since the $\Ww_5$ minimal
model has $c_{mm}=\ft87 \ge1$, that there will be an infinite number of
descendant operators.

\section{Hierarchies of string embeddings\label{sct:embeddings}}

     It was proposed recently \cite{vafa} that as part of the general
programme of looking for unifying principles in string theory, one should
look for ways in which string theories with smaller worldsheet symmetries
could be embedded into string theories with larger symmetries.  In
particular, it was shown in \cite{vafa} that the bosonic string could be
embedded in the $N=1$ superstring, and that in turn, the $N=1$ string
could be embedded in the $N=2$ superstring.  In subsequent papers, it was
shown by various methods that the cohomologies of the resulting theories were
precisely those of the embedded theories themselves \cite{Jose:universalstring,kato:N=01embedding}.

     The essential ingredient in the embeddings discussed in \cite{vafa} is
that a realisation for the currents of the more symmetric theory can be found
in terms of the currents of the less symmetric theory, together with some
additional matter fields whose eventual r\^ole for the cohomology is to
supply degrees of freedom that are cancelled by the additional ghosts of the
larger theory.  For example, the $N=1$ superconformal algebra, at critical
central charge $c=15$, can be realised in terms of a $c=26$ \emt\ $T_M$
as\col \bea
T &=& T_M-\ft32 b_1\, \del c_1 -\ft12 \del b_1\, c_1 +\ft12 \del^2(c_1\,
\del c_1)\nonumber\\
G &=& b_1 +c_1\, (T_M + \del c_1\, b_1) +\ft52 \del^2 c_1\,,
\label{N10}
\eea
where $b_1$ and $c_1$ are ghost-like dimension $(\ft32,-\ft12)$
anticommuting matter fields. The cohomology of the BRST operator for the
$N=1$ superstring, with this realisation of the $N=1$ superconformal
algebra, is precisely that of the usual bosonic string
\cite{vafa,Jose:universalstring,kato:N=01embedding}.  This is most easily
seen using the method of \cite{kato:N=01embedding}, where a unitary
canonical transformation $Q\longrightarrow e^{R}\, Q\, e^{-R}$ is applied
to the $N=1$ BRST operator, transforming it into the BRST operator for the
bosonic string plus a purely topological BRST operator.  In effect, the
degrees of freedom of $b_1$ and $c_1$ are cancelled out by the degrees of
freedom of the commuting dimension $(\ft32,-\ft12)$ ghosts for the
dimension $\ft32$ current $G$. The central charge of the \emt\ for
$(b_1,c_1)$ is $c=11$, which precisely cancels the $c=-11$ central charge
for the dimension $\ft32$ ghost system for the dimension $\ft32$ current
$G$.

     It is natural to enquire whether some analogous sequence of embeddings
for \WS s might exist, with, for example, the usual Virasoro string
contained within the $\Ww_3$--string, which in turn is contained in the
$\Ww_4$--string, and so on \cite{vafa}. In fact, as was observed in
\cite{TAM:zhao1}, such sequences of embeddings are already well known for
\WS s. The simplest example is provided by the $\Ww_3$--string,
where the $\Ww_3$--currents $T$ and $W$ are realised in terms of an
\emt\ $T_X$, and a scalar field $\varphi$. The $\varphi$ field here plays
a role analogous to the $(b_1,c_1)$ matter fields in the embedding of
the bosonic string in the $N=1$ superstring. Here, however, the central
charge $c=\ft{149}2$ for the \emt\ of $\varphi$ does not quite cancel the
central charge $c=-74$ of the $(\beta, \gamma)$ ghosts for the dimension
$3$ current $W$, and so the nilpotence of the $\Ww_3$ BRST operator
requires that $T_X$ has central charge $c=\ft{51}2$ rather than $c=26$.
The $\varphi$ field has no associated continuous degrees of freedom in
physical states, and the cohomology of the $\Ww_3$ string is just that of
a $c=25\ft12$ Virasoro string tensored with the Ising model.

     It has also been suggested that one might be able to embed the $c=26$
Virasoro string into, for example, the $\Ww_3$ string.  However, it would,
perhaps, be surprising if it were possible to embed the Virasoro string into
the $\Ww_3$ string in two different ways, both for $c_X=\ft{51}2$ and also for
$c_X=26$. Indeed, there is no known way of realising the currents of the
$\Ww_3$ algebra, with the central charge $c=100$ needed for nilpotence of the
BRST operator, in terms of a $c=26$ \emt\ plus other fields
that would contribute no continuous degrees of freedom in physical states.

\begin{intermezzo}
A very different approach was proposed in \cite{bfw}, where it was shown
that by performing a sequence of canonical transformations on the BRST
operator of the $\Ww_3$ string, it could be transformed into the BRST
operator of an ordinary $c=26$ bosonic string plus a purely topological
BRST operator. However, as was shown in \cite{TAM:zhao2}, and
subsequently reiterated in \cite{west}, one step in the sequence of
canonical transformations involved a nonlocal transformation that
reduced the original $\Ww_3$ BRST operator to one with completely trivial
cohomology. A later step in the sequence then involved another nonlocal
transformation that caused the usual cohomology of the bosonic string to
grow out of the previous trivial cohomology. In effect one is glueing two
trivialised theories back to back, and so the physical spectra of the two
theories prior to trivialisation are disconnected from one another, making
the embedding quite meaningless.
\end{intermezzo}

     An interesting possibility for generalising the ideas in \cite{vafa} is
to consider the case where the bosonic string is embedded in a fermionic
higher-spin string theory.  The simplest such example would be provided by
looking at a theory with a dimension $\ft52$ current in addition to the
\emt.  In order to present some results on this example,
it is useful first to recast the $N=1$ superstring, with the matter currents
realised as in (\ref{N10}), in a simpler form.  We do this by performing a
canonical redefinition involving the dimension $2$ ghosts $(b,c)$, the
dimension $\ft32$ ghosts $(r,s)$, and the ghost-like matter fields
$(b_1,c_1)$ (which we shall refer to as pseudo-ghosts).  If we transform
these according to\col
\bea
c &\longrightarrow& c-s\, c_1 \nonumber\\
r &\longrightarrow& r-b\, c_1\nonumber\\
b_1 &\longrightarrow& b_1+b\, s\,,\label{vtrans}
\eea
(with $b$, $s$ and $c_1$ suffering no transformation), then the BRST
operator assumes the graded form $Q=Q_0+Q_1$, where\col
\bea
Q_0 \!&=&\! \oint c\Big( T_M +T_{b_1c_1} +T_{rs} +\ft12 T_{bc} + x\,
\del^2(\del c_1 c_1) \Big)\nonu
Q_1 \!&=&\! \oint s \Big( b_1 -x\, b_1 \del c_1 c_1 +3x\, r \del s c_1
+ x\, \del r s c_1 +2x^2\, \del^2 c_1 \del c_1 c_1\Big)\,.
\label{redefBRST}
\eea
Here $x$ is a free constant which actually takes the value $-\ft12$
when one transforms (\ref{N10}) according to (\ref{vtrans}), but can be made
arbitrary by performing a constant OPE-preserving rescaling of $b_1$ and
$c_1$.  The reason for introducing $x$ is that it can be viewed as a
power-counting parameter for a second grading of $Q_0$ and $Q_1$, under the
$(b_1,c_1)$ pseudo-ghost number.  Thus $Q_0$ has terms of pseudo-ghost
degrees 0 and 2, whilst $Q_1$ has terms of pseudo-ghost degrees $-1$, 1 and
3. (We have dropped an overall $x^{-1}$ factor from $Q_1$ for convenience.
We are free to do this owing to the first grading under $(r,s)$ degree,
which implies that $Q_0^2=Q_1^2=\{Q_0,Q_1\}=0$.)

     Before moving on to the generalisation to higher dimensions, it is useful to
present the unitary canonical transformation of ref.\ \cite{kato:N=01embedding} in this
language, which maps the BRST operator into that of the bosonic string plus
a topological term.  Thus we find that the charge\col
\be
R=\oint c_1 \Big(-c\, \del r -\ft32 \del c\, r -x\, r\, s\, \del c_1\Big)
\ee
acts on the BRST operator $Q=Q_0+Q_1$ to give\col
\be
e^R\, Q\, e^{-R} = \oint c\, (T_M -b\, \del c) +\oint s\, b_1\,.
\label{eq:embedtransform}
\ee
The first term on the \rhs\ is the usual BRST operator of the bosonic
string, and the second term is purely topological, with no
cohomology.

     We may now seek a dimension $(2,\ft52)$ generalisation of this
dimension $(2,\ft32)$ theory.  Thus we now consider commuting ghosts
$(r,s)$ of dimensions $(\ft52,-\ft32)$ for a dimension $\ft52$ current, and
anticommuting pseudo-ghosts $(b_1,c_1)$ of dimensions $(\ft52,-\ft32)$.  We
find that a graded BRST operator $Q=Q_0+Q_1$ again exists, where $Q_0$
contains terms with pseudo-ghost degrees 0, 2 and 4, whilst $Q_1$ has terms
of pseudo-ghost degrees $-1$, 1, 3 and 5.  The coefficients of the various
possible structures in $Q_0$ and $Q_1$ are determined by the nilpotency
conditions $Q_0^2=Q_1^2=\{Q_0,Q_1\}=0$.  $Q_0$ takes the form\col
\bea
\lefteqn{Q_0=\oint c\, \Big( T_M +T_{b_1c_1}+T_{rs} +\ft12 T_{bc} + x\,
\del^2\big(3\del^3 c_1 c_1 +7 \del^2 c_1 \del c_1\big)}
\hspace*{10em}\nonumber\\
&& +\ y\ \del^2\big( \del^3 c_1 \del^2 c_1 \del c_1 c_1\big)\Big)\ ,
\label{5Q0}
\eea
where $x$ and $y$ are arbitrary constants associated with the terms in $Q_0$
of pseudo-ghost degree 2 and 4 respectively.  The form of $Q_1$ is quite
complicated\col
\bea
\lefteqn{Q_1=\oint s\, \Big( b_1 - 6x\, b_1 \del^2 c_1 \del c_1 -4x\, b_1
\del^3 c_1 c_1 -6x\, \del b_1 \del^2 c_1 c_1 -}
\hspace*{8em}\nonumber\\
&&2x\, \del^2 b_1 \del c_1 c_1 +\cdots\nonumber\\
&&+  x\, \big(\ft{26}{3} x^2 +\ft{25}{6} y\big) \del^4 c_1
\del^3 c_1 \del^2 c_1 \del c_1 c_1 \Big)\ ,
\label{5Q1}
\eea
where the ellipsis represents 13 terms of pseudo-ghost degree 3.

     One may again look for a charge $R$ that acts unitarily and canonically
on the BRST operator to give it a simpler form.  We find that the required
charge is given by\col
\bea
\leqn{R=\oint c_1\, \Big( -c \del r -\ft52 \del c r -x\, c\del^2 c_1
\del c_1  \del r - \ft52 x\, \del c \del^2 c_1 \del c_1 r}
& & - 2x\, \del c_1 \del^2 r s -6x\, \del^2 c_1 \del
r s + 2x\, \del^3 c_1 r s -\ft12 y\, \del^3 c_1 \del^2 c_1 \del
c_1 r s\Big)\ .
\eea
Acting on the BRST operator $Q=Q_0+Q_1$, this gives exactly
\vgl{eq:embedtransform}, which shows that this theory is again simply
equivalent to the bosonic string.

     Although the dimension $(2,\ft52)$ theory that we have described above has a
BRST operator that is a natural generalisation of the $N=1$ superconformal
BRST operator with the realisation (\ref{N10}) for the matter currents,
there is one important aspect which we should consider.  From the graded
$(2,\ft32)$ BRST operator given by (\ref{redefBRST}), one can invert the
canonical transformation (\ref{vtrans}), and get back to a form in which one
can replace the specific realisation (\ref{N10}) of the superconformal
currents by an abstract realisation in terms of currents $T$ and $G$.  In
this sense, one can say that the realisation (\ref{N10}) describes an
embedding of the bosonic string in the $N=1$ superstring.
Let us look if the some procedure goes through for the $(2,\ft52)$
BRST operator.


Let us consider the
$\Ww_{2,5/2}$ algebra in more detail.  Classically, the primary
dimension $\ft52$ current $G$ satisfies the OPE\col
\be
G(z)G(w) \sim {T^2\over z-w}\ .
\ee
The Jacobi identity is satisfied modulo the classical null field\col
\be
N_1\equiv 4T\, \del G -5 \del T\, G \ .
\ee
Before specifying what we mean with a classical null field, we wish to
show that one can realise the algebra in the following way:
\bea
T&=& -\ft12 \psi\, \del\bar\psi +\ft12 \del\psi\, \bar\psi \ ,\nonumber\\
G&=& \ft12\left(\psi+ \bar\psi\right)\, T \ ,
\eea
where $\psi$ is a complex fermion satisfying the OPE $\psi(z)\bar\psi(w)
\sim 1/(z-w)$.  One can easily verify for this realisation that the null
field $H$ vanishes.

We checked that both for the abstract algebra, and for the realisation no
BRST operator can be constructed along the familiar lines, \eg by
introducing ghosts--antighost pairs for $T$ and $G$, and constructing a
BRST charge $c^1 T+  c^2 G + f_{ij}{}^k c^ic^jb_k$, where the structure
constants $f_{ij}{}^k$ depend on $T$. To explain this surprising fact, we
need to elaborate on the meaning of a null field in a Poisson algebra.

We use exactly the same definition for a null field as in the OPE case,
see subsection \ref{ssct:OPAs}. That is, all null fields form an ideal in
the Poisson algebra. We can check by repeatedly computing Poisson brackets
with $N_1$ that it there is an ideal in the Poisson algebra of $T$ and
$G$, generated by $N_1$ and
\be
N_2 \equiv 4 T^3 - 30 \dz G\, G\ .
\ee
More precisely, all other null fields can be written as\col
\be
f_1(T,G) N_1 + f_2(T,G) N_2\ ,
\ee
with $f_i(T,G)$ a differential polynomial in $T$ and $G$. We see that the
phase space of the Poisson algebra is not simply the space of differential
polynomials in $T$ and $G$, but the additional constraints $N_1=N_2=0$
have to be taken into account. In such a case, the ordinary procedure of
constructing a (classical) BRST charge does not work. Indeed, one should
use the BRST-formalism appropriate for reducible constraints, which
requires the introduction of ``ghosts for ghosts'', see \eg
\cite{Henneaux}. This clearly explains why no ``ordinary'' BRST charge
exists for this system.

Thus, it seems that the $\Ww_{2,\ft52}$ string is of very different type
than other strings considered up to now. It remains to be seen if the
resulting BRST-charge is in any way related to the one we constructed
above, eqs,~(\ref{5Q0},\ref{5Q1}).

We have
explicitly checked for all higher half-integer dimensions, and we find
that again a classical $\Ww_{2,n/2}$ algebra does not identically satisfy
the Jacobi identity.  Thus again, we expect that ghosts for ghosts should
be introduced to enable a proper treatment in the BRST formalism of such
algebras.

%The above situation is very different from that for the integer-dimension
%$\Ww_{2,s}$ strings.  In that case the theories were again originally
%constructed by generalising the graded BRST operator structure
%$Q=Q_0+Q_1$ of the $\Ww_3=\Ww_{2,3}$ string to arbitrary $s$
%\cite{TAM:hs}, and a priori one had no particular reason to expect
%that the resulting BRST operators could be viewed as describing string
%theories with dimension $2$ and dimension $s$ currents. It was really
%only by using arguments of the kind presented in section \ref{sct:W2s},
%where we showed that the $\Ww_{2,s}$ BRST operators describe the
%quantisation of classical theories with classical $\Ww_{2,s}$ algebras as
%local symmetries, that their identification as string theories could be
%made.


\section{Conclusion and discussion}
    In this chapter we have looked at the quantisation of $\Ww$-string
theories based on the classical \w2s higher-spin algebras.  One of the
more noteworthy features of these theories is that anomaly-free quantisation
is possible even when there does not exist a closed quantum extension of the
classical $\Ww_{2,s}$ algebra at the critical central charge.
We can identify quantum currents as the coefficients of the dimension
$-1$ and $1-s$ ghosts in the BRST current, and discarding antighosts.
Of course, these fields do generate a realisation of a quantum \WA.
However, the corresponding abstract algebra is probably infinitely
generated.  A previous example of this kind of phenomenon,
where a BRST operator exists even when the matter system does not
generate a closed algebra at the quantum level, was found in the context
of the non-critical $\Ww_3$--string discussed in
\cite{BSS:noncritstring,BLNW:noncritW3,wolfB}.


It is quite puzzling that there can be several inequivalent quantum
theories that arise from the same classical theory, corresponding to
different possible choices for the coefficients of the quantum
corrections to the classical BRST operator.
A study of the relation between the classical and quantum cohomology of
the corresponding BRST operators should be able to shed some light on this
point.
\puzzle{classical
cohomology bekijken (maar wanneer)}

     In a multi-scalar realisation, the spectrum of physical states for a
$\Ww_{2,s}$ string turns out to be described by the tensor product of sets
of bosonic-string states in the effective spacetime times certain primary
operators built from the $(\varphi, \, \beta, \, \gamma)$ fields. In most
cases these primary fields are conjectured to correspond to those of some
Virasoro or $\Ww$ minimal model. For example, the regular sequence of
$\Ww_{2,s}$ BRST operators, which exist for all $s$, corresponds to the
lowest unitary $\Ww_{s-1}$ minimal model, with $c_{mm}=2(s-2)/(s+1)$. We
have tested the above conjecture in detail for the cases $s=4$, 5 and 6
of the \w2s--string. We have shown, for the lowest few levels, that
indeed the operators $U(\varphi, \beta, \gamma)$ that arise in the
physical states, are associated with the highest weight fields of the lowest unitary
$\Ww_{s-1}$ minimal models. Specifically, we find in all physical states
that $U(\varphi, \beta, \gamma)$ is either a highest weight field of the corresponding
$\Ww_{s-1}$ minimal model, or else it is a descendant field in the sense
that it is obtained from a highest weight field by acting with the negative modes of the
primary currents of the $\Ww_{s-1}$ algebra. Since the central charge
$c=\ft{2(s-2)}{(s+1)}$ of the $(\varphi, \beta, \gamma)$ system satisfies
$c\ge1$ for $s\ge5$, it follows that in these cases there are infinite
numbers of such descendant fields in the models. Thus the $\Ww_{s-1}$
generators provide a strikingly powerful organising symmetry in these
cases.

The original realisations of the $\Ww_N$ algebras were the $(N-1)$-scalar
realisations from the Miura transformation, introduced in
\cite{fazam:W3freefield,faly:WAn}. By contrast, the realisations of the
lowest unitary $\Ww_{s-1}$ minimal models that we find here are all given
in terms of just one scalar field $\varphi$, and the $(\beta, \gamma)$
ghost system for dimension $s$. This ghost system can be bosonised, yielding
two-scalar realisations. However even when $s=4$, our two-scalar
realisation is quite different from the usual Miura realisation of $\Ww_3$.
In particular, our realisations close on the $\Ww_{s-1}$ algebras modulo
the appearance of certain null primary fields in the OPEs of the currents,
whereas no such null fields arise in the Miura realisations. Presumably
the realisations that we find here are very specific to the particular
unitary minimal models that arise in these higher-spin string theories.
As an example, we present the dimension $2$ and dimension $3$ currents
\vgl{eq:WTmm} and \re{eq:W4spin3} for the $\Ww_3$ algebra at $c=\ft45$ in
the bosonised language, where $\gamma=\exp({i\rho})$ and
$\beta=\exp({-i\rho})$\col \bea
T_{mm}&=&-\ft12 (\del\varphi)^2 -\ft12 (\del\rho)^2 -\alpha\, \del^2\varphi
+\ft72 i\, \del^2\rho,\nonu
W_{mm}&=&\sqrt{\ft2{13}}\Big( \ft53\,(\del\varphi)^3 +5\alpha\,
\del^2\varphi\, \del\varphi +\ft{25}4\, \del^3\varphi +4\,
\del\varphi\, (\del\rho)^2 -16i\, \del\varphi\, \del^2\rho \nonu
&&\qquad -12i\, \del^2\varphi\, \del\rho -\ft23 i\alpha\, (\del\rho)^3
-3\alpha\, \del^2\rho\, \del\rho -\ft{11}6 i\alpha\, \del^3\rho\Big)\,,
\eea
where $\alpha^2=\ft{243}{20}$.  It is interesting to note that this
realisation of the $\Ww_3$ algebra at $c=\ft45$ is precisely the one
obtained in \cite{Gron:W3real} (case $I$, after an $SO(1,1)$ rotation of
the two scalars), where more general scalar realisations of $\Ww_3$ modulo
a null dimension $4$ operator were considered.

If there is just one $X^\mu$ coordinate in the effective \emt\ $T_X$, the
spectrum of physical states for the \w2s string becomes more complicated, as
observed in \cite{TAM:hs}. In particular, there are additional physical
states over and above those of the form \re{eq:Wphysop}, which do not
factorise into the product of effective-spacetime physical states times
operators $U(\varphi, \beta, \gamma)$. Examples of these were found for the
$\Ww_3$ string in \cite{TAM:W3string}, and for \w2s strings in
\cite{TAM:hs}. A general discussion of the BRST cohomology for the
two-scalar $\Ww_3$ string is given in \cite{BCP:W3cohom}. It may well be
that the \w2s strings with just one additional coordinate $X^\mu$ capture
the more subtle aspects of the underlying higher-spin geometry.

     We have looked also at string theories based on
classical algebras involving a higher-spin fermionic current in addition to
the \emt.  These classical algebras do not satisfy the Jacobi identity
identically, but only modulo null fields.  When there exists a classical
realisation, these null fields are identically zero. The appearance of
(classical) null fields obliges one to introduce ``ghosts for ghosts''.
This topic remains to be studied furher.

%12345678901234567890123456789012345678901234567890123456789012345678901234567890
%J_z=J_+ = \k/2 \dz g g^{-1},A_-=A_\zb,\del_-=\db

\chapter[\WA s and gauged WZNW models]{Extensions of the Virasoro algebra
and gauged \WZW s\label{chp:WZNW}}
One of the most general methods to obtain a realisation of an extended
Virasoro algebra is Drinfeld-Sokolov reduction \cite{HRdrisok,bais}.
This method consists of imposing certain constraints on the currents of a
\KA\ $\bar{g}$. The constraints are associated to an embedding of $\slt$
in the Lie algebra $g$. In the classical case, the reduced phase space
then forms a realisation of a \WA\ \cite{HRdrisok,bais}, as reviewed in
section \ref{sct:classicalHR}.

In \cite{bais}, it was shown that the constraints reduce the classical
WZNW Ward identities to those of the extended Virasoro algebra. This
points towards a connection between the induced actions of the \KA\ and
the \WA. This connection was made explicit in the case of $\Ww_3$ in
\cite{ZFacW3dbg}. In section \ref{sct:quantumHR} this is generalised to
arbitrary (super) \KA s \cite{us1}. The main idea is to implement the
constraints on a gauged \WZW. This leads to a path integral formulation
in the Batalin-Vilkovisky formalism of the induced action of the \WA\ in
terms of the WZNW action.

To complete the construction, we show in section \ref{sct:cohomology}
that the quantum currents of the constrained \WZW\ form a realisation of
a quantum \WA. These results are used in section \ref{sct:RNSBVq} to find
the quantum corrections to the Batalin-Vilkovisky action.

The $\asp$ affine Lie algebra will be used as an example in sections
\ref{sct:OSPpart1} and \ref{sct:OSPpart2}.  The $\slt$ embedding we use,
gives a realisation of the $N$-extended $so(N)$ superconformal algebras
\cite{NLN=3kniz,NLN=3bersh}.

This chapter contains results published in \cite{us1,BerkProcRN}. Several
results presented here are new. For conventions on \WZW s, we refer to
section \ref{sct:WZW}. Appendix
\ref{app:super} fixes our conventions for (super) Lie algebras and summation
indices.



\section{Classical Drinfeld-Sokolov reduction\label{sct:classicalHR}}
In this section we will briefly explain the classical Hamiltonian
reduction that gives a realisation of a (classical) \WA\ in terms of
a \KA. References \cite{HRdrisok,bais,classHRFORTW,susyHRFRS,D:thesis} 
can be consulted for further details.


Consider a \KA\ $\hat{g}$ with level $\k$, with a (super) Lie algebra $\bar{g}$
valued field $J_z(z)=J^a_z(z)\ {}_at$, where $t_a$ are matrices representing
$\bar{g}$. For a given $\slt$-embedding, we impose a set of
constraints on the currents $J_z^a$\col
\bea
\P_-\left(J_z- \frac \k 2 e_-\right) = 0\,.
\label{eq:constraints1}
\eea
These constraints are all first class in the terminology of Dirac
\cite{DiracConstraints}, except $\P_{-1/2}J_z=0$. It is more convenient
to have all constraints first class. For a (bosonic) Lie algebra, this
can be achieved by taking only half of the constraints at grading $-1/2$.
It is possible \cite{classHRFORTW} to do this by choosing an alternative integral
grading. However, this procedure does not work for most
super Lie algebras. Therefore we will introduce auxiliary fields instead. We
postpone the discussion of this case with half-integral $\slt$ grading to
the next section. So we ignore in this section any sign issues for
the super case.

In the reduced phase space, there is a gauge freedom because elements are
considered equivalent modulo the constraints.
The infinitesimal gauge transformations \re{eq:WZWPB}\col
\bea
\delta_\eta J_z(x) &=&
-\int \Big\{\str \left(\eta(y) J_z(y)\right), J_z(x)\Big\}_{\rm PB}\nonu
&=&{\k\over 2} \del \eta(x) + [\eta(x),J_z(x)],\qquad\qquad \eta\in\P_+\bar{g}
\label{eq:JgaugeTransf}
\eea
can be used to impose suitable gauge conditions. Several choices are
possible, but we will mainly use the highest weight gauge\col
\bea
\Big(1-\PHW\Big)\left(J_z- \frac \k 2 e_-\right) = 0\,.
\label{eq:constraintsHWG}
\eea

We will now compute the reduced Poisson algebra in the highest weight
gauge. This can be done in several ways. First, one can view the gauge
conditions as an extra set of constraints. Because the constraints are then
second class, Dirac brackets should be used. However, this approach
requires inverting the matrix of the Poisson brackets of the constraints,
which is in general quite difficult.

A second way is by using gauge invariant polynomials. These are
polynomials in the currents which are gauge invariant under
\vgl{eq:JgaugeTransf} up to the constraints \re{eq:constraints1}. It can
be shown \cite{Dublin:HRgaugeinvpol} that for each highest weight
current, there is a gauge invariant polynomial
\footnote{See appendix \ref{app:super} for the basis for
the generators of the super Lie algebra.}\col
\be
\tilde{J}_z^{(j, \alpha_j)} = J_z^{(j j, \alpha_j)}\Bigr|
  \raisebox{-1.5ex}{constraints \vgl{eq:constraintsHWG}}
\label{eq:gaugeinvpol}\,.
\ee
The $\tilde{J}_z^{(j, \alpha_j)}$ are unique modulo the constraints
\re{eq:constraints1}. They generate all gauge invariant polynomials by using
addition and multiplication. The polynomials \re{eq:gaugeinvpol} can be
computed by finding the unique (finite) gauge transformation which brings
the constrained current \vgl{eq:constraints1} to the gauge fixed form
\vgl{eq:constraintsHWG}. This is used in \cite{D:thesis,DSST:string,S:thesis}
to construct an algorithm to obtain the polynomials
$\tilde{J}_z^{(j, \alpha_j)}$.\\
Because the gauge transformations are compatible with the Poisson
brackets, the gauge invariant polynomials form a (nonlinear) subalgebra
with respect to the original Poisson bracket. Moreover, we can see that
this algebra is isomorphic to the reduced Poisson algebra.
\begin{proof}
To show that the Poisson algebra of the gauge invariant polynomials is
isomorphic to the reduced Poisson algebra, we observe that their Dirac
and Poisson brackets are the same (modulo \re{eq:constraints1}). Recall
the definition of the Dirac bracket for some second class constraints
$\chi^\alpha$\col
\beastar
\leqn{\{F(x),G(y)\}_D = \{F(x),G(y)\}_{\rm PB} -}
 &&\int d^2x_0d^2y_0 \{F(x),\chi^\alpha(x_0)\}_{\rm PB}
   \Delta_{\alpha\beta}(x_0,y_0) \{\chi^\beta(y_0),G(y)\}_{\rm PB}\,,
\eeastar
with $\Delta_{\alpha\beta}$ the inverse matrix of $\{\chi^\alpha,
\chi^\beta\}_{\rm PB}$. When $F(x)$ is gauge invariant, \ie $\{F,
\chi^\alpha\}_{\rm PB}$ is zero modulo constraints, we see that
$\{F, G\}_D = \{F,G\}_{\rm PB}$ modulo constraints.
\end{proof}

A third way to compute the reduced phase space is to use cohomological
techniques \cite{KO:BRST}. For every constraint
$\chi^\alpha$ in \re{eq:constraints1}, we introduce a ghost $c_\alpha$
and antighost $b^\alpha$ pair, to which we assign ghost number $+1$ and
$-1$ respectively. We define the Poisson brackets\col
\be
\{b^\alpha(x), c_\beta (y)\}_{\rm PB} =
\delta^\alpha{}_\beta\ \delta^{2}(x-y)\,.
\label{eq:PBbc}
\ee
The BRST operator $Q$ acting in the complex $\ca$ generated by
$\{J^a_z, b^\alpha\}$ is defined by\col
\bea
Q(A)(x) = \int d^2y\ \{{\cal J}(y),A(x)\}_{\rm PB}\,,&&A\in\widehat{\ca}\,,
\label{eq:BRSTdef}
\eea
where the ghost number $+1$ BRST current\col
\be
{\cal J} =  c_\alpha(x) \ {}^\alpha\chi(x) +
\mbox{higher order in ghostfields}\,,
\ee
is determined by requiring that $Q^2=0$. Because the constraints
\re{eq:constraints1} form a linear Poisson algebra, we know that this extra
part contains only ghosts. We now study the cohomology of $Q$\col
\be
{\cal H}^*(Q;\widehat{\ca}) = {{\rm ker}(Q) \over {\rm im}(Q)}\,,
\ee
\ie we look for fields annihilated by $Q$, but as $Q^2=0$, we
use the equivalence relation\col
\be
A\sim A +Q(B)\,.    \label{eq:BRSTequivalence}
\ee
Let us see how this is related to the constraints. $Q$ on a field $f$,
which does not contain $b$ or $c$, gives the gauge transformation
\re{eq:JgaugeTransf} with parameter $c$ of $f$. Obviously, we are only
interested in fields modulo the gauge transformations, \ie modulo ${\rm
im}(Q)$. Now consider a ghost number zero functional of the currents and
(anti)ghosts $F[J, b, c]$. We will call $f[J]\equiv F[J, b=0, c=0]$.
Because $Q$ acts as a derivation, the BRST transformation of $F$ is
schematically given by\col
\be
Q(F) = Q(f) + G_\alpha Q(b^\alpha) + H^\alpha Q(c_\alpha) + Q(J) I\,,
\ee
where $G, H, I$ are other functionals with $I$ depending on $b, c$.
Because of the Poisson brackets \re{eq:PBbc} this gives\col
\be
Q(F) = \delta_c(f) + G_\alpha \chi^\alpha +\ {\rm terms\ with\ antighosts}\,,
\ee
where $\delta_c$ denotes the gauge transformation \re{eq:JgaugeTransf}
with parameter $c$. This means that if $Q(F)=0$, its ghost--free part $f$
is gauge invariant modulo constraints. Moreover, it can be shown that the
``dressing'' of a gauge invariant polynomial to an element of the
cohomology is unique, see \eg\ \cite{Henneaux}. Hence, computing the
cohomology of $Q$ corresponds to finding the gauge invariant polynomials,
and the gauge choice \re{eq:constraintsHWG} corresponds to a choice of
representative for an equivalence class \re{eq:BRSTequivalence}. In
\cite{dBT:WAandLA}, an iterative method was provided to construct the
generators of the cohomology for the case of bosonic Lie algebras. We
will discuss the generalisation of this method in section
\ref{sct:cohomology}.\\
Quantisation of the Drinfeld-Sokolov reduction using a BRST approach was
initiated in \cite{quanHRBO}. The general case for bosonic \KA s, was
treated in \cite{dBT:WAandLA}. We will treat superalgebras in section
\ref{sct:cohomology}.


Having characterised the reduced phase space, we now give some general
comments about its structure. The current corresponding to $e_+$ (which
has $j=1$) is common to all reductions (as we are treating $\slt$
embeddings). It turns out that it satisfies a Virasoro Poisson bracket.
This is still the case if we add the Sugawara tensor $T_S^{(0)}$ for the \KA\
formed by the highest weight $j=0$ currents. We define\col
\be
T_s = 2 y\tilde{J}_z^{(1,0)} + s T_S^{(0)}\,,\label{eq:RNTsclassdef}
\ee
which is a Virasoro current for $s=0,1$. With respect to $T_1$, all the other
currents $\tilde{J}_z^{(j, \alpha_j)}$ are primary and have spin $j+1$
\cite{classHRFORTW}, see also section \ref{sct:cohomology}. So, we have
found a representation of a classical \WA\ for any $\slt$ embedding in a
(super) Lie algebra. See \cite{FRS:sl2embed} for a classification of all
embeddings and the dimensions of the generators of the resulting \WA.

\section{Quantum reduction\label{sct:quantumHR}}
In this section, the previous scheme for constraining the currents of
a \KA\ is implemented by gauging a \WZW. This will enable us to quantise
the reduction and to give a path integral expression for the induced
action of the \WA\ which results from the construction.

\subsection{Gauged WZNW Model\label{ssct:constrgaugedWZW}}
The affine Lie algebra $\hat{g}$ is realised by a \WZW\ with action $\k
S^-[g]$. This action has the global symmetry \re{eq:WZWdeltaJ}. To make
part of the symmetries local, we introduce gauge fields $A_z\in\P_-\bar{g}$
and  $A_\zb\in\P_+\bar{g}$ and the action\col
\bea
{\cal S}_0&=&\k S^-[g]-
\frac{\k}{2\pi x} \int \str\left( A_z\,g^{-1}\bdel g\right) +
\frac{\k}{2 \pi x}\int\str \left(A_{\zb}\del g g^{-1}\right) +\nonu
&&\frac{\k}{2 \pi x}\int \str \left(A_{\zb}g A_zg^{-1}\right)\,.\label{eq:RNaction1}
\eea
Parametrising the gauge fields as $A_z\equiv \del h_-\, h_-^{-1}$ and
$A_{\zb}\equiv\bdel h_+\, h_+^{-1}$ where $h_\pm\in \P_\pm\bar{g}$, we
see that $\cs_0$ is invariant under\col
\bea
h_\pm &\rightarrow&\g_\pm h_\pm\nonu
g&\rightarrow&\g_+ g \g_-^{-1},\label{eq:RNgauget1}
\eea
where
\be
\g_\pm\in\P_\pm\bar{g}.
\ee
This can be shown by using the Polyakov-Wiegman formula \re{eq:RNpwfor} to
bring \vgl{eq:RNaction1} in the form ${\cal S}_0=\k S^-[h_+^{-1}gh_-]$ and
noting that $S^-[h_\pm]=0$. We will always work in the gauge $A_z=0$ and
hence drop all contributions of this gauge field.

In order to impose the constraints \re{eq:constraints1}, we
would like to add the following term to the action\col
\be
\cs_{\rm extra}=-\frac{\k}{2 \pi x} \int \str \left( A_\zb  e_{-} \right),
\ee
using the gauge field as a Lagrange multiplier. The variation of this term
under the gauge transformations \vgl{eq:RNgauget1} is\col
\be
\delta \cs_{\rm extra} = -{\k\over 2\pi x}\int\str\left([\eta,A_\zb]e_-\right).
\ee
The variation is nonzero when $\P_{1/2}\bar{g}$ is not empty. We would like to
preserve the gauge invariance. As already mentioned, this can be done for
bosonic algebras by choosing an integral grading. This is used in
\cite{dBT:WAandLA} to quantise the Drinfeld-Sokolov reduction for bosonic
algebras. However, for a super Lie algebra, another method has to be
used. We restore gauge invariance by introducing the ``auxiliary" field
$\t\in\P_{1/2}\bar{g}$ and define\col
\bea
{\cal S}_1&=&\k S^-[g]+ \frac{1}{\p x} \int \str\left( A_\zb\left( J_z
-\frac \k 2 e_- -\frac \k 2 [\t,e_-]\right)\right)\nonu
&&+ \frac{\k}{4\pi x}\int \str \left( [\t,e_-]\db\t\right),
\label{eq:RNBactiongen}
\eea
with the affine currents \re{eq:WZWJgroupclassical} $J_z=\frac \k 2 \dz
g g^{-1}$. If $\t$ transforms under \re{eq:RNgauget1} as\col
\be
\t\rightarrow\t+\P_{1/2}\eta\,,
\label{eq:RNgaugettau}
\ee
where $\exp \eta\equiv \gamma^+$, the action $\cs_1$ turns out to be
invariant. To prove this we need\col
\be
str\left([\eta,A_\zb][\t,e_-]\right)=0\,,
\ee
which follows from $\P_{-1/2}[\eta,A_\zb]=0$.

The Lagrange multipliers $A_\zb$ impose now the constraints\col
\be
\P_- J_z - \frac \k 2 e_- -\frac \k 2 [\t,e_-]\ =\ 0\,.
\label{eq:fullconstraints}
\ee
Using the gauge symmetry we can further reduce the current by choosing
$\P_{\geq 0}J_z\in\KER{+}$ and $\t=0$, thus indeed reproducing the
highest weight gauge \vgl{eq:constraintsHWG}.

As discussed in the previous section, the polynomials in the affine
currents and $\tau$ which are gauge invariant modulo the constraints, \ie the
equations of motion of $A_\zb$, form a realisation of a classical extended
Virasoro algebra. The \WA\ is generated by the polynomials $\tilde{J}_z^{(j,
\alpha_j)}$ \re{eq:gaugeinvpol}. Of course, we can take a different set of
generators. For example, it is customary to use the Virasoro operator $T_1$
\re{eq:RNTsclassdef}. We will denote the new generators as
$\ct^{(j, \alpha_j)}$, and assemble them for convenience in a matrix\col
\be
\ct\equiv \ct^{(j, \alpha_j)}\ {}_{(jj, \alpha_j)}t\in\KER{+}\,.
\ee

\subsection{The induced action\label{ssct:RNindact}}
We couple the generators $\ct$ of the \WA\ to sources $\mu\in\KER{-}$
and add this term to the action ${\cal S}_1$ \re{eq:RNBactiongen}\col
\bea
{\cal S}_2={\cal S}_1+\frac{1}{4\p x y}\int \str\left(\mu \ct\right)\,.
\label{eq:RNBS2def}
\eea
However, $\ct$ is only invariant up to the constraints
\re{eq:fullconstraints}, \ie equations of motions of the gauge fields
$A_\zb$. To make $\cs_2$ gauge invariant, we modify the
transformation rules of the $A_\zb$ with $\mu$-dependent terms. As the
gauge fields appear linearly in the action, no further modifications are
needed. The action is then invariant under $\P_+\bar{g}$ gauge
transformations. Due to the new transformation rules of $A_\zb$, the
gauge algebra now closes only on-shell.

Provided that at quantum level the currents $\ct^{(j,\alpha_j)}$, up to
multiplicative renormalisations and normal ordering,
satisfy a quantum version of the \WA, we obtain a realisation of the
induced action for this \WA\col
\be
\exp\left( -\Gamma[\mu]\right)=\int [\d g\, g^{-1}][d\t][d A_{\zb}]\left(
\mbox{Vol}\left( \P_+\bar{g} \right) \right)^{-1}
\exp\left(-\cs_2[g,\t,A_\zb,\mu]\right)\,.\label{eq:RNokok}
\ee
Of course, we should properly define the above expression. The gauge
fixing procedure is most easily performed using the
Ba\-ta\-lin--Vil\-kov\-isky (BV) method \cite{bv}. We skip the details
here, a readable account of the BV method can be found in \eg
\cite{Frank:thesis,BVKUL,BVWT,proeyen}. For every generator of the gauge
algebra, we introduce a ghost field $c^a$ with statistic $(-1)^{1+a}$,
$c\in\P_+\bar{g}$. Furthermore, for every field we introduce an
antifield of opposite statistics, $J_z^*\in\bar{g}$,
$A_\zb^*\in\P_-\bar{g}$, $\t^*\in\P_{-1/2}\bar{g}$ and
$c^*\in\P_-\bar{g}$. We denote all fields $\{J_z, \t, A_\zb, c\}$
collectively with $\Phi^\alpha$ with corresponding antifields
$\Phi_\alpha^*$. The first step of the BV method consists of extending
the action $\cs_2$\col \be
\cs_{\rm BV} =
\cs_2 +
 {1\over \pi}\int \Phi_\alpha^* \delta_c\Phi^\alpha+\ldots\,,
\label{eq:RNBVguess}
\ee
where $\delta_c$ denotes the gauge transformations with the parameter
$\eta$ replaced by the ghost $c$. The ellipsis denotes extra terms at
least quadratic in the ghosts and antifields such that the classical BV
master equation is satisfied\col
\be
(\cs_{\rm BV},\cs_{\rm BV}) = 0\,,\label{eq:BVmaster}
\ee
where the bracket denotes the BV antibracket\col
\be
(A,B)\equiv \int d^2x\left(
   {\stackrel{\leftarrow}{\delta} A \over \delta\Phi^\alpha(x)}
       {\stackrel{\rightarrow}{\delta} B \over \delta_\alpha\!\Phi^*(x)} -
   {\stackrel{\leftarrow}{\delta} A \over \delta\Phi^*_\alpha(x)}
       {\stackrel{\rightarrow}{\delta} B \over \delta^\alpha\!\Phi(x)}
   \right)\,.
\ee
The master equation is guaranteed to have a solution by closure of the
gauge algebra \cite{BV:classmastereq,ULB:classmastereq,Henneaux,toine}.
Note that the master equation \re{eq:BVmaster}, together with
Jacobi identities for the antibracket, implies that the operator $Q_{\rm
BV}$ defined by\col
\be
Q_{\rm BV}(A)\ =\ (\cs_{\rm BV}, A)\label{eq:RNQBV}
\ee
is nilpotent. This is provides a link to the BRST operator. In particular,
for gauge algebras which close without using the equations of motion, this
definition coincides with \vgl{eq:BRSTdef} for $Q$ acting on fields.

We find that the solution to the master equation is given by\col
\bea
\cs_{\rm BV}&=&\cs_1+\frac{1}{2\pi x}\int \str\bigg(
   - c^*cc + J^*_z\left(\frac \k 2 \dz c + [c,J_z] \right) + \t^*c\nonu
&& +\, A^*_\zb\left(\db c + [c,A_\zb]\right)
       +{1\over 2y}\mu\, \hat{\ct}\bigg)\,,
\label{eq:RNBbvbv}
\eea
where $\mu_{(j, \alpha_j)}$ appears linearly in the extended action.
$\hat{\ct}^{(j, \alpha_j)}$ reduces to the gauge invariant polynomials
$\ct^{(j, \alpha_j)}$, when antifields and ghosts are set to zero,
see \cite{DSST:string,S:thesis}. We will determine the exact form of
$\hat{\ct}^{(j, \alpha_j)}$ at the end of this section.

The gauge is fixed in the BV method by performing a canonical
transformation and by putting the new antifields to zero. A canonical
transformation between fields and antifields leaves the antibracket
invariant. One should use a canonical transformation such that the
new fields have a well-defined propagator. \\
When going to the quantum theory, a quantum term has to be added to the BV
master equation \re{eq:BVmaster} in order to ensure that the results of
the theory are gauge-in\-depen\-dent. This implies that quantum corrections
have to be added to the extended action. We will not make a fully
regularised quantum field theory computation, as this does not seem feasible
for the general case. Instead, we use BRST invariance as a guide. We will
use OPE-techni\-ques without specifying a regularisation underlying this
method in renormalised perturbation theory. The OPEs will enable us to
present results to all orders in the coupling constant $\k$.

To be able to use OPEs, we should choose a gauge where we can assign
definite OPEs to the fields. We will put $A_\zb=0$, which is different
from the highest weight gauge. In the BV formalism, this amounts to using
a canonical transformation which simply interchanges $A^*_\zb$ and
$A_\zb$. Moreover, we rename the old antigauge fields $A^*_\zb$ to
BRST-antighosts $b$. The gauge-fixed action reads\col
\bea
{\cal S}_{\rm gf}&=&\k S^-[g]
+ \frac{\k}{4\pi x}\int \str \left( [\t,e_-]\db\t \right)
+\frac{1}{2\pi x}\int \str\left( b\db c\right)\nonu
&&+\ \frac{1}{4\pi x y} \int\str \left( \mu\hat{\ct}\right)\,.
\label{eq:RNBactiongen2}
\eea

\begin{intermezzo}\label{int:RNFFOPEs}%
Using the results of section \ref{sct:FreeFields}, we see that the
fields satisfy the OPEs\col
\bea
\tau^a \tau^b &=&\ope{{2\over\k} {}^ah^b}\nonumber\\
b^a c^b&=& \ope{{}^a g^b}\,,
\label{eq:RNOPEs}
\eea
where ${}^ah^b$ is the inverse of\col
\be
{}_ah_b = (e_-)^c \ {}_{ca}f^d\ {}_dg_b\,.
\ee
The currents $J_z^a$ satisfy the \KA\ $\hat{g}$ with level $\k$ (in the
classical and quantum case). The tensors $g$ and $f$ are defined in
appendix \ref{app:super}.
\end{intermezzo}

Corresponding to the nilpotent operator \vgl{eq:RNQBV}, we have the BRST
charge\footnote{Note that $\{b,c\}$ denotes an anticommutator,
not a Poisson bracket.}\col \bea
Q=\frac{1}{4\p i x}\oint \str\left( c \left( J_z -\frac \k 2
e_--\frac \k 2[\t,e_-]+ \frac 1 4 \{b,c\}\right) \right)\,.
\label{eq:RNBQdef}
\eea
It is easy to see that indeed $Q(\Phi^\alpha) = \delta_c \Phi^\alpha$.

We now comment on the explicit form of the currents $\hat{\ct}^{(j,\alpha_j)}$.
BRST invariance of the action requires $\hat{\ct}$ to be BRST invariant.
This determines $\hat{\ct}^{(j, \alpha_j)}$ up to BRST exact pieces. This
will lead us to the study of the BRST cohomology in section
\ref{sct:cohomology}. In particular, we will prove in subsection
\ref{ssct:RNclasscohomology} that the classical cohomology is generated by
the gauge invariant polynomials $\tilde{J}_z^{(j, \alpha_j)}$ after
replacing the current $J_z$ with the
``total current'' $\hat{J}_z^{(j, \alpha_j)}$\col
\be
\hat{J}_z\equiv J_z +{1\over 2}\{b,c\}\,.\label{eq:RNJhat}
\ee
This means that in the gauge $A_\zb=0$, $\hat{\ct}^{(j, \alpha_j)}$ is
given by performing the same substitution in the invariant polynomials
$\ct^{(j, \alpha_j)}$. This result determines $\hat{\ct}^j$ in the
extended action \re{eq:RNBbvbv}, \ie before putting antifields to
zero\col
\be
\hat{\ct}[J_z,\tau,A_\zb^*,c]\ =\ \ct[J_z +{1\over 2}\{A_\zb^*,c\},\tau]\,,
\label{eq:That}
\ee
where we indicated the functional dependence of the gauge invariant
polynomials as $\ct[J_z, \tau]$. An elegant argument for this formula purely
relying on BV methods can be found in \cite{S:thesis}.

We can now check, using the classical OPEs given in
intermezzo \ref{int:RNFFOPEs}, that the currents $\P_+\hat{J}_z$ defined
in \re{eq:RNJhat} satisfy the same Poisson brackets as $\P_+ J_z$. Because
$\ct$ is a functional of only the positively graded currents (and $\tau$),
the polynomials $\hat{\ct}^{(j, \alpha_j)}$ still form a representation of
the \WA.

We are now in a position to go to the quantum theory. BRST invariance
will be our guideline. We notice that the expression \vgl{eq:RNBQdef} for
the BRST charge has no normal ordering ambiguities. We will take it as
the definition for the quantum BRST operator, and we will determine any
quantum corrections to extended action by requiring invariance under the
quantum BRST transformations. For $\mu=0$, the gauge fixed action
\re{eq:RNBactiongen2} is quantum BRST invariant. Hence, any
corrections for the gauge fixed action reside only in the $\hat{\ct}^{(j,
\alpha_j)}$. The quantum corrections to the generators of the classical
\WA\ will be determined in section \ref{sct:cohomology} by studying
the quantum cohomology.

\section{An example : $\asp$\label{sct:OSPpart1}}
In this section we present an explicit example to make our treatment
of the classical case more concrete.

The Lie algebra $\asp$ is generated by a set of bosonic generators\col
\be
t_{\pp}, t_0, t_=, t_{ab},\qquad t_{ab}=-t_{ba} \mbox{ and } a, b\in\{1,
\cdots, N\}\,,
\ee
which form an $\slt\times \soN$ Lie algebra and a set of
fermionic generators\col
\be
t_{+a}, t_{-a},\qquad a\in\{1,\cdots, N\}\,.
\ee
We will represent a Lie algebra valued field $X$ by $X^{\pp}\
{}_{\pp}t+X^{0}\ {}_0t+X^{=}\ {}_=t + \frac{1}{2} X^{ab}\ {}_{ab}t +
X^{+a}\ {}_{+a}t + X^{-a}\ {}_{-a}t$, where the matrices ${}_at$ are in
the fundamental representation (which has index $x=1/2$)\col
\be
X \equiv \left( \begin{array}{ccc} X^0 & X^{\pp} & X^{+b} \\
        X^= & -X^0 & X^{-b} \\
        X^{-a} & -X^{+a} & X^{ab}
\end{array} \right)\,,
\label{eq:RNBospmat}
\ee
and $X^{ab}=-X^{ba}$.
From this, one reads the generators of $\asp$ in the fundamental
representation and one can easily compute the (anti)com\-mu\-ta\-tion
relations. The dual Coxeter number for this algebra is
$\tilde{h}={1\over 2}(4-N)$. In the fundamental representation, the
supertrace \re{eq:strdef} of two fields is\col
\bea
\str (X\, Y)&=&2 \,X^0\,Y^0 + X^{\pp}\,Y^{=} + X^{=}\,Y^{\pp} -
       X^{ab}\,Y^{ba} +\nonu
&&  (-1)^Y \,2\,X^{-a} Y^{+a} - (-1)^Y\, 2\,X^{+a} Y^{-a} \,,
\label{eq:ospstr}
\eea
with $(-1)^Y$ a phase factor depending on the parity of $Y$, which is
$(-1)$ for (anti)ghosts and $(+1)$ for all other fields. In this equation
we inserted the phase factor explicitly. In this section, and in section
\ref{sct:OSPpart2}, the summation convention is applied without
introducing extra signs. Also, indices are raised and lowered using the
Kronecker delta.

\begin{intermezzo}\label{int:Osp OPEs}%
We list the $\asp$ classical OPEs for convenience\col
\[
\begin{array}{l@{\!}ll@{\!}l}
J_z^0J_z^0&=\ \opetwo{\frac \kappa 8}{0}&
   J_z^{i}J_z^{j}&=\ \opetwo{\frac \kappa 8 \delta^{ij}}{-\frac{
        \sqrt{2}}{4} f^{ij}{}_kJ_z^k}\\
J_z^{\pp}J_z^=&=\ \opetwo{\frac \kappa 4}{J_z^0}&
   J_z^{i}J_z^{\pm a}&=\
       \ope{\frac{\sqrt{2}}{4}\lambda_{ab}{}^iJ_z^{\pm b}}\\
J_z^0J_z^\ppmm&=\ \ope{\pm \frac 1 2 J_z^\ppmm}&
   J_z^{\pm a}J_z^{\pm b}\,&=\ \ope{\mp \frac 1 4 \delta^{ab}J_z^\ppmm}\\
J_z^0J_z^{\pm a}&=\ \ope{\pm \frac 1 4 J_z^{\pm a}}\hspace*{1em}&
J_z^\ppmm J_z^{\mp a}&=\ \ope{\frac 1 2 J_z^{\pm a}}\\[2mm]
J_z^{+a}J_z^{-b}&
\multicolumn{3}{l}{  =\ \opetwo{\frac \kappa 8 \delta^{ab}}{\frac 1 4
     \left(\delta^{ab}J_z^0+\sqrt{2}\lambda_{ab}^iJ_z^i\right)}\,,
}
\end{array}
\]
where the index $i$ stands for a pair of indices $(pq)$ with $1\leq p < q
\leq N$, and $\delta^{(pq)(rs)}=\delta^{pr}\delta^{qs}$.
The second order ``poles'' are given by $-\k/2\
{}^ag^b$ where the metric agrees with \vgl{eq:ospstr}. The normalisations
are such that $\lambda_{ab}{}^{(pq)}\equiv 1/\sqrt{2}(\d_a^p\d_b^q -
\d_a^q\d_b^p)$, ${[}\lambda^i,\lambda^j{]}=f^{ij}{}_k\lambda^k$, ${\em
tr}(\lambda^i\lambda^j)=-\d^{ij}$, $f^{ik}{}_l f^{jl}{}_k=-(N-2)\d^{ij}$ where
$f^{ij}{}_k$ are the structure constants of the $\soN$ subalgebra.
Note that the classical and quantum OPEs are the same.
\end{intermezzo}

We consider the embedding corresponding to the $\slt$ subalgebra
formed by $\{t_{\pp}, t_0, t_=\}$. It has index of embedding
\re{eq:RNindexydef} $y=1$. The constraints \vgl{eq:fullconstraints} are
in our example simply\footnote{ $\tau^{+a}$ changed sign with respect to
\cite{us1} because $[\tau, e_-] = \tau_{\rm \cite{us1}}$.}\col
\bea
J_z^= = {\kappa\over 2} &&J_z^{-a} = -{\kappa\over 2}\tau^{+a}\, .
\eea

The elements of the classical extended Virasoro algebra are
polynomials which are invariant (up to equations of motion of the
gauge fields) under the $\P_+\osp$ gauge transformations. The generators
are found by carrying out the unique (finite) $\P_+\osp$ gauge
transformation which brings the currents $J_z$ and $\tau$ in the highest
weight gauge, \ie $\tilde{J}_z^0 = \tilde{\t}^{+a}=0$ for the transformed
currents. We find\col
\bea
\tilde{J}_z^{\pp}&=&J_z^{\pp}+\frac 2 \k J_z^0 J_z^0+2 J_z^{+a}\t^{+a}-
   \sqrt{2}\l_{ab}{}^iJ_z^i\t^{+a}\t^{+b}
   -\del J_z^0-\frac \k 2 \del \t^{+a} \t^{+a},\nonu
\tilde{J}_z^{+a}&=&J_z^{+a} +\sqrt{2}\l_{ab}{}^iJ_z^i \t^{+b}+
   J_z^0\t^{+a}-\frac \k 2\del\t^{+a}\nonu
\tilde{J}_z^{i}&=&J_z^i+\frac{\k}{2\sqrt{2}}  \l_{ab}{}^i\t^{+a}\t^{+b}
\,. \label{eq:RNBpolys}
\eea
We refer to \cite{us1} for the gauge transformations of these polynomials,
and give only one example\col
\bea
\d \tilde{J}_z^{i}& =&
  -\frac{\p}{2\sqrt{2}}\l_{ab}{}^i\h^{+a}\frac{\d{\cal
    S}_1}{\d A^{+b}_\zb}\,.\label{eq:RNBnoninvar}
\eea
Computing the Poisson brackets of the generators, we find that the
operator $T_s$ \re{eq:RNTsclassdef} is given by\col
\bea
T_s =2\left(\tilde{J}_z^{\pp} +
   s {2\over \k} {\tilde{J}_z^{i}}{\tilde{J}_z^{i}}\right)\, .
\label{eq:RNBTsdef}
\eea
It satisfies the Virasoro Poisson brackets for $s=0$ and $1$ (with
central charge $-6\k$). The latter choice is to be preferred as the other
generators $\tilde{J}_z^j$ are then primary fields \cite{classHRFORTW}.
We can now readily identify\col
\be
T\equiv T_1,\qquad G^a\equiv 4i\, \tilde{J}_z^{+a}, \qquad
U^i=-2 \sqrt{2}\tilde{J}_z^{i}\,,\label{eq:RNSONgeneratorsclass}
\ee
as the generators of the classical $N$-extended $\soN$ superconformal
algebra, with the level $k$ of the $so(N)$ currents given by $k=-2\k$,
see intermezzo \ref{int:SO(N)superOPEs}.

\begin{intermezzo}\label{int:SO(N)superOPEs}%
The $N$-extended $\soN$ superconformal algebras 
\cite{NLN=3kniz,NLN=3bersh} 
are generated by the \emt\ $T$, $N$ dimension 3/2
supersymmetry currents $G^a$ and affine $\soN$ currents $U^{i}$.
For $N=1$ and $N=2$ these are just the standard $N=1$ and $N=2$
superconformal algebras. For $N\geq 3$, the algebra is nonlinearly
generated. The subalgebra of transformations globally defined on the
sphere, form an $\asp$ algebra. The nontrivial (classical and
quantum) OPEs are given by\col
\bea
G^a G^b &=& \opethree{\d^{ab}\beta}{\frac{\beta}{k} \l_{ab}{}^i U^{i}}
  {2 \d^{ab}T+\frac{\beta}{2k} \l_{ab}{}^i\del U^{i}+
  \gamma\P_{ab}^{ij}(U^iU^j)}\nonu
U^{i}U^{j}&=&\opetwo{-\frac k 2\d^{ij}}{f^{ij}{}_kU^k}\nonu
U^{i}G^a&=&\ope{\l_{ba}{}^iG^b},\label{eq:RNSO(N)algebra}
\eea
where $k$ is the level of the $\soN$ \KA. $\l_{ab}{}^i$ and $f^{ij}{}_k$
are defined in intermezzo \ref{int:Osp OPEs}, and
$\P_{ab}^{ij}=\P_{ba}^{ij}=\P_{ab}^{ji} = \l_{ac}{}^{i}\l_{cb}{}^{j} +
\l_{ac}{}^{j}\l_{cb}{}^{i} + \d_{ab}\d^{ij} $. The only difference between the
classical and quantum OPEs of the generators is in the constants $c, \beta$
and $\gamma$. Although we need only the classical OPEs here, we give the
expressions for the quantum case to avoid repetition\col
\[
c\ =\ \frac k 2 \frac{6k + N^2-10}{k+N-3}\hspace*{2em}
\beta\ =\ k\frac{2k+N-4}{k+N-3}\hspace*{2em}
\gamma\ =\ \frac{2}{k+N-3}\,.
\]
The values of $c,\beta$ and $\gamma$ for classical OPEs are given by the
large $k$ limit of these expressions.
\end{intermezzo}

We choose for the generators $T^{(j, \alpha_j)}$ of the $so(N)$
superconformal algebra the fields appearing in
\vgl{eq:RNSONgeneratorsclass} $T_s, G^a,U^i$, keeping $s$ arbitrary.
We couple these
generators to sources $h, \psi_a, A_i$ and add this term to the action
$\cs_1$ as in \vgl{eq:RNBS2def}\col
\bea
\cs_2 = \cs_1 + {1\over \pi } \int \, h T_s + \psi_a G^a + A_i U^i\,,
\eea
To preserve the gauge invariance of the resulting action, we have to
modify the transformation of the gauge fields $A_\zb$ such that the
equation of motion terms in (\ref{eq:RNBnoninvar}) are canceled. This
reflects itself in the terms proportional to the antifields $A^*_\zb$ in
the Batalin-Vilkovisky action. Because the new gauge algebra closes only
on-shell, we need terms quadratic in the antifields to find an extended
action satisfying the BV master equation. The result is (see
\vgl{eq:RNBbvbv}, and eq.\ (4.20) of \cite{us1}) \footnote{We use a more
convenient normalisation for the antifields here compared to \cite{us1}.
We have $(A^{+a*}_\zb) =-{\pi\over2}(A^{+a*}_\zb)_{\rm
\cite{us1}}$ and
$(A^{\pp\, *}_\zb) =\pi (A^{\pp\, *})_\zb)_{\rm
\cite{us1}}$.}\col
\bea
\leqn{\cs_{\rm BV}\ =\
\mbox{ (terms independent of $h, \psi_a, A_i$})}
&& +{1\over\pi}\int A_\zb^{\pp\, *}\left( -c^{\pp}\del h -
   \frac 4 \k c^{\pp}h J_z^0-2i c^{\pp}\j_a\t^{+a} \right)\nonu
&&-{2\over\pi}\int A_\zb^{+b *}\left(-\frac 1 2 c^{+b}\del h +
   c^{\pp}h\t^{+b} -\frac 2 \k c^{+b}h J_z^0 +
   s \frac{2\sqrt{2}}{\k}\l_{ab}{}^i c^{+a} J_z^ih\right.\nonu
&&\qquad - ic^{\pp} \j_b +i c^{+a} \j_a \t^{+b}-i c^{+b} \j_a
  \t^{+a}-i c^{+a} \j_b \t^{+a} -\l_{ab}{}^i c^{+a} A_i\Biggr)\nonu
&&- \frac{2}{\p\k}\int A^{\pp\, *}_\zb A^{+a*}_\zb c^{+a}c^{\pp}h+
  s \frac{1}{\p\k}\int A^{+a*}_\zb A^{+a*}_\zb
  c^{+b}c^{+b}h\,.\label{eq:RNBbvresult}
\eea
We now fix the gauge with the condition $A_\zb=0$. Renaming the
anti-gauge fields into antighosts\col
\bea
A^{+a*}_\zb =b^{-a}, && A^{\pp\, *}_\zb =b^= \,,
\eea
we end up with the gauge-fixed action \vgl{eq:RNBactiongen2}\col
\bea
{\cs}_{\rm gf}&=&\k S^-[g]-
  \frac \k \p\int \t^{+a}\bdel\t^{+a}+
  \frac{1}{\p}\int b^=\bdel c^{\pp}-
  \frac 2 \p\int b^{-a}\bdel \g^{+a}+\nonu
&&\frac 1 \p\int\left( h \hat{T_s}+\j_a \hat{G}^a+ A_i\hat{U}^i \right)\,,
\label{eq:RNBgauges1}
\eea
where $\hat{T}_s$, $\hat{G}_a$ and $\hat{U}_i$ have precisely the same
form as in eqs.\ \re{eq:RNBTsdef} and \re{eq:RNSONgeneratorsclass}, but
the currents $J$ are replaced by their hatted counterparts
\re{eq:RNJhat}\col
\bea
\hat{J}_z^{\pp}&=&J_z^{\pp} \nonu
\hat{J}_z^{+a}&=&J_z^{+a} -\frac 1 2 c^{\pp}b^{-a}\nonu
\hat{J}_z^{0} &=&J_z^{0} -\frac 1 2 b^= c^{\pp}+ \frac 1 2 b^{-a}\g^{+a}\nonu
\hat{J}_z^{i} &=&J_z^{i} + \frac{1}{\sqrt{2}} \l_{ab}{}^i b^{-a} \g^{+b}\,.
\label{eq:RNBcurrent3}
\eea
This substitution rule is valid independent of the value of $s$ one
takes. The hatted generators are classically invariant under the action
of the BRST-charge \vgl{eq:RNBQdef}.

\section{Cohomology \label{sct:cohomology}}
In \cite{us1}, the quantum cohomology of the BRST
operator \re{eq:RNBQdef} was studied. Ref.~\cite{us3} completely
characterised the cohomology. This section summarises the
results and extends the study to the classical case. In addition we prove
that the construction method as given in \re{eq:RNBQdef} is not unique.

When treating the quantum case, all products of fields are considered
regularised using point splitting, and normal ordering is performed from
right to left. Where algebraic expressions are involved -- like when taking
the supertrace or the commutator of two fields -- it is always understood
that the fields are not reordered\col
\be
[X,Y]\equiv -(-1)^{aY}[X^aY^b ]_0 \ {}_{ba} f^c\ {}_ct\,.
\ee

\subsection{Computing the quantum cohomology\label{ssct:RNquantcohomology}}
Consider the OPA $\ca$ generated by the basic fields $\{b, \hat{J}_z, \t,
c\}$. In the quantum case the currents $\P_{\geq 0}\hat{J}_z$
satisfy the same OPEs as $\P_{\geq 0}J_z$, except for the central
extension. For two currents of zero $\slt$ grading we find
\footnote{In \cite{us3} this formula was claimed to be
true also when $\hat{J}_z^a$ and $\hat{J}_z^b$ do not have zero $\slt$
grading. This is not correct as such an OPE involves explicit
(anti)ghosts. However, these OPEs are never needed in the computation of
the cohomology, so the results of \cite{us3} are not influenced.\puzzle{}}
\col
\be
\hat{J}_z^a \hat{J}_z^b =
    \opetwo{-{\k\over 2}\ {}^ag^b +
            (-1)^{\underline{c}}\
            {}_{\underline{c}}f^{ad}\ {}_d f^{b\underline{c}}\ }
           {\ \hat{J}_z^c\ {}_cf^{ab}}\,,
\ee
where an index $\underline{c}$ is limited to generators of strictly
negative grading. Similarly, we will use $\overline{c}$ for an index
restrained to strictly positive grading.

To every field $\Phi$, we assign a double grading
$[\Phi]=(k,l)$, where $k\in\frac 1 2 \Bbb{Z}$ is the $\slt$ grading and
$k+l\in \Bbb{Z}$ is the ghost number. The ``auxiliary'' fields $\t$
are assigned the grading $(0,0)$. The algebra $\ca$ acquires a double
grading\col
\bea
\ca=\bigoplus_{\stackrel{\scriptstyle m,n\in\frac 1 2{\bf
  Z}}{m+n\in{\bf Z}}}\ca_{(m,n)}
\eea
and OPEs preserve the grading. The BRST charge \re{eq:RNBQdef} decomposes
into three parts of definite grading, $Q=Q_0+Q_1+Q_2$, with
$[Q_0]=(1,0)$, $[Q_1]=(\frac 1 2,\frac 1 2)$ and $[Q_2]=(0,1)$\col
\bea
Q_0&=&-\frac{\k}{8\pi i x}\oint \str \left(c e_-\right)\nonu
Q_1&=&-\frac{\k}{8\pi i x}\oint \str \left(c\left[\t,e_-\right]\right)\,.
\label{eq:RNBqsqr} \eea
As illustrated in fig.\ \ref{fig:BRSgrading}, the operators $Q_0$,
$Q_1$ and $Q_2$, map $\ca_{(m,n)}$ to $\ca_{(m+1,n)}$,
$\ca_{(m+\frac 1 2 ,n+ \frac 1 2)}$ and $\ca_{(m,n+1)}$ respectively.
\begin{figure}[ht]
\centering\portpict{dc1}(9,6.3)
\caption{$Q_0$, $Q_1$ and $Q_2$ acting on $\ca_{(\frac 1 2 ,\frac 3
2)}$.\label{fig:BRSgrading}}
\end{figure}

It follows from $Q^2=0$ that
$Q_0^2=Q_2^2=\{Q_0,Q_1\}=\{Q_1,Q_2\}=Q_1^2+\{Q_0,Q_2\}=0$, but\col
\be
Q_1^2=-\{Q_0,Q_2\}=\frac{\k}{32 \pi i x}\oint \str\left( c \left[
\P_{1/2}c,e_-\right] \right)\label{eq:RNBQ12}
\ee
does not vanish. The presence of $Q_1$ is the main difference with the
case of bosonic \KA s treated in \cite{dBT:WAandLA}, as the auxiliary field
$\tau$ was not introduced there. The action of $Q_0$,
$Q_1$ and $Q_2$ on the basic fields is given in table
\ref{table:RNBqonf}. Note that due to \vgl{eq:OPEJacRAB}, the BRST
charges $Q_i$ act as derivations on a normal ordered product of fields.
\begin{table}[hbt]
\begin{center}
\small
\newlength{\tmp}
\setlength{\tmp}{1em}
\def\tmptab#1{\makebox[\tmp][l]{$ #1$} $\rightarrow $\ }
\begin{tabular}{|l|l|l|}
\hline
$\hfil Q_0$ &$\hfil Q_1$ &$\hfil Q_2$\\*[1mm]
\hline
\tmptab{b}$ -\frac \k 2 e_- $&
\tmptab{b}$ -\frac \k 2 [\t,e_-] $&
\tmptab{b}$ \P_-\hat{J}_z$\\*[1mm]
\tmptab{c}$ 0$&
\tmptab{c}$ 0$&
\tmptab{c}$ \frac 1 2 cc  $\\*[1mm]
\tmptab{\hat{J}_z}$ -\frac \k 4 [e_-,c]$&
\tmptab{\hat{J}_z}$ -\frac \k 4 [[\t,e_-],c]$&
\tmptab{\hat{J}_z}$ \frac 1 2 [c,\P_{\geq
       0}\hat{J}_z]+\frac \k 4 \dz c $\\*[1mm]
 & &$\hspace*{3em}-{1\over 2}\dz c^a\ {}_af^{\underline{b}\overline{c}}\
                  {}_{\overline{c}\underline{b}}f^d\ {}_dt $\\*[1mm]
\tmptab{\t}$ 0$&
\tmptab{\t}$ \frac 1 2 \P_{+1/2} c$&
\tmptab{\t}$ 0$\\*[1mm]
\hline
\end{tabular}
\end{center}
\caption{The action of $Q_i$ \label{table:RNBqonf}}
\end{table}

The subcomplex $\ca^{(1)}$, generated by $\{ b,\
\P_-\hat{J}_z-\frac \k 2 [ \t,e_-]\}$ has a trivial cohomology
${\cal H}^*(\ca^{(1)};Q)=\Bbb{C}$. Moreover, its elements do not appear in the
action of the charges $Q_i$ on the other fields. One can show that
the cohomology of $\ca$ is then equal to the cohomology of the reduced
complex $\wha=\ca / \ca^{(1)}$, generated by $\{\P_{\geq 0} \hat{J}_z, \t,
c\}$\col\ ${\cal H}^*(\ca; Q)={\cal H}^*(\wha; Q)$. Because $\wha$
contains only positively graded operators, the OPEs close on the reduced
complex. Obviously the double grading on $\ca$ is inherited by $\wha$.
\begin{figure}
\centering\portpict{dc2}(9,6.5)
\caption{$\wha$ and its filtration $\wha^n$.\label{fig:filtration}}
\end{figure}
At this point, the theory of spectral sequences \cite{bott} is applied to
compute the cohomology on $\wha$. We summarise the
results. Details can be found in \cite{us3}. The filtration
$\wha^m$, $m\in\frac 1 2 {\bf Z}$ of $\wha$ (see fig.
\ref{fig:filtration})\col
\bea
\wha^m\equiv \bigoplus_{k\in\frac 1 2 {\bf Z}}\bigoplus_{l\geq
m}\wha_{(k,l)}\,,
\eea
leads to a spectral sequence $E_r={\cal H}^*(E_{r-1};d_{r-1})$, $r\geq1$
which converges  to ${\cal H}^*(\wha;Q)$. The sequence starts with
$E_0=\wha, d_0=Q_0$. The derivation $d_r$ represents the action of $Q$ at
that level. One shows that $E_1= {\cal H}^*(\wha ;Q_0) \simeq\wha
\left[\PHW \hat{J}_z \right]\otimes\wha \left[ \t\right]\otimes\wha
\left[ \P_{\frac 1 2 }c\right]$, where $\wha \left[\Phi \right]$ denotes the
subalgebra of $\wha$ generated by $\Phi$. The spectral sequence collapses
after the next step and we find\col
\bea
{\cal H}^*(\ca;Q)\simeq E_2={\cal H}^*(E_{1};Q_1)=
\wha \left[\PHW \left(\hat{J}_z +\frac\k
     4[\t,[e_-,\t]]\right)\right]\,.
\eea
This result means that the dimension of the cohomology equals the
number $n_j$ of $sl(2)$ irreducible representations in the branching of the
adjoint representation of $\bar{g}$. Furthermore, the generators of the
cohomology have ghost number zero. Because the reduced complex has no
antighosts, the generators consist only of the currents $\P_{\geq
0}\hat{J}_z$ and $\tau$.

We now outline a recursive procedure to obtain explicitly the
generators of the cohomology\col\ the tic-tac-toe construction
\cite{bott} (see fig.\ \ref{fig:tictactoe}).
The generators of the cohomology are split up as\col
\bea
W^{(j,\alpha_j)}=\sum_{r=0}^{2j}W^{(j,\alpha_j)}_r\,,
\label{eq:RNBlterm}
\eea
where $W^{(j,\alpha_j)}_r$ has grading $(j-\frac r 2 , -j+\frac r
2)$. The leading term $W^{(j,\alpha_j)}_0$ is given by\col
\bea
W^{(j,\alpha_j)}_0= \hat{J}_z^{(jj,\alpha_j)}+\d_{j,0}\frac
\k 4 [\t ,[e_-,\t]]^{(00,\alpha_j)} \label{eq:RNBlterm0}
\eea
and the remaining terms are recursively determined by\col
\bea
Q_0W^{(j,\alpha_j)}_r=-Q_1 W^{(j,\alpha_j)}_{r-1} - Q_2
W^{(j,\alpha_j)}_{r-2}\,,\label{eq:RNtictactoe}
\eea
where $W^{(j,\alpha_j)}_r=0$ for $r<0$ or $r>2j$. As an example, one
can check that for $j> 1/2$\col
\be
W^{(j,\alpha_j)}_1=-[\tau,\P_{\geq 0}\hat{J}]^{(jj,\alpha_j)}\,.
\ee
For $j=1/2$ this equation gets a correction term that depends only on
$\tau$ such that \vgl{eq:RNtictactoe} is satisfied, see \cite{us3}, and
also \vgl{eq:RNospcohogenqua}.

\begin{figure}
\centering\portpict{dc3}(9.5,6)
\caption{The tic-tac-toe construction for a conformal dimension 2
current.\label{fig:tictactoe}}
\end{figure}
There is a certain ambiguity in this construction because any combination
of the generators $W^{(j, \alpha_j)}$ is still BRST invariant (and
non-trivial). This corresponds to the fact that $Q_0$ annihilates
$\hat{J}_z^{(jj, \alpha_j)}$, and thus one can freely add at each step in
the iteration \re{eq:RNtictactoe} any combination of correct grading of
these currents to $W^{(j, \alpha_j)}_r$. We will use the notation
$W^{(j, \alpha_j)}$ for the elements of the cohomology which reduce to a
single highest weight current when discarding non-highest weight currents,
more precisely\col
\be
W^{(j,\alpha_j)} = \hat{J}_z^{(j,\alpha_j)}\Bigr|
  \raisebox{-1.5ex}{$(\P_+ - \PHW)\hat{J}\ =\ \tau\ =\ 0$}\,.
  \label{eq:RNWAgendef}
\ee

An important step for practical applications of this construction, is
the ``inversion" of $Q_0$ in \re{eq:RNtictactoe}. Table \ref{table:RNBqonf}
shows that it is necessary to find a field $X$ such that $Q_0(X)=\P_{\geq
1}c$. To find this field, we introduce an operator $L$
\cite{dBGFH:Walginfclass} in $\bar{g}$, which is defined by\col
\be
\left\{ \begin{array}{ll}
L(_at) = 0,&               {\rm if }\ {}_at\in \PHW \bar{g}\\[2mm]
{[}e_-, L(_at)]\ =\ {}_at,&{\rm otherwise\,.}
\end{array}\right.
\label{eq:RNLdef}
\ee
Using table \ref{table:RNBqonf} we see that\col
\be
Q_0(L\hat{J}_z)={\k\over 4}\P_{\geq 1}c\,.
\ee
We will need $L$ in subsection \ref{ssct:RNWA} to construct $W^{(1,0)}$.

%\subsection{Miura transformation}
%The (quantum) Miura transformation is given by the map $W \rightarrow
%\P_0W$ \cite{tjin1,us3}. To show that this map is an algebra
%isomorphism, $\P_0W^{(j,\alpha_j)}$ has to be nonzero, and their OPEs
%have to close. The latter follows easily because the OPEs preserve the
%grading. To prove that they do not vanish, we notice that
%$\P_0W^{(j,\alpha_j)}$ is equal to the last term in the expansion
%\vgl{eq:RNBlterm}. Consider now the mirror spectral sequence $E_r'={\cal
%H}^*(E_{r-1}';d_{r-1}')$, starting at $E_0=\wha, d_0=Q_2$, associated to the
%filtration\col
%\bea
%\wha^m{}'\equiv \bigoplus_{l\in\frac 1 2 {\bf Z}}\bigoplus_{k\geq
%m}\wha_{(k,l)} \,.
%\eea
%Using table \ref{table:RNBqonf}, one shows that $E_1'$ is only nonvanishing
%at grading $(\frac m 2 ,\frac m 2)$, $m\geq 0$. As we already know that
%$E_\infty '$ for this spectral sequence vanishes unless the ghostnumber is
%zero, we find that $W^{(j,\alpha_j)}_{2j}$ is indeed the last,
%nonvanishing term in \vgl{eq:RNBlterm}.

\subsection{Classical cohomology\label{ssct:RNclasscohomology}}
Before discussing the extended Virasoro algebra in this cohomology,
let us see how these results are modified when computing the
classical cohomology.

The action of $Q_i$ on the fields, table \ref{table:RNBqonf}, remains the
same except for $Q_2(\hat{J}_z)$. Here the last and most complicated term
disappears as it originates from double contractions. This means that
exactly the same reasoning can be followed as in the quantum case. The
cohomology can be computed in the reduced complex $\wha$ generated by
$\{\P_{\geq 0}\hat{J}_z, \tau, c\}$. Because the generators
$W^{(j, \alpha_j)}$ have ghost number zero, we see that any element of
the cohomology can be written in terms of $\{\P_{\geq 0}\hat{J}_z,
\tau\}$.

On the other hand, we know from general arguments that the
elements from the cohomology (at ghostnumber zero) reduce to
gauge invariant polynomials when $b$ and $c$ are set to zero (see the
discussion at the end of section \ref{sct:classicalHR}).
We conclude that the elements of the classical cohomology can be found by
substituting $J_z$ with $\hat{J}_z$ in the gauge invariant polynomials.
In particular, it follows from eqs.\ \re{eq:gaugeinvpol} and
\re{eq:RNWAgendef} that \col
\be
W^{(j, \alpha_j)}_{\mbox{\small classical}} = \tilde{J}_z^{(j, \alpha_j)}\Bigr|
  \raisebox{-1.5ex}{$J_z\rightarrow \hat{J}_z$} \,.
\ee
This result was used in subsection \ref{ssct:RNindact} to determine the
terms that couple to the sources $\mu$ in the classical extended action
\re{eq:RNBbvbv}.

\subsection{The quantum \WA\label{ssct:RNWA}}
The generators $W^{(j, \alpha_j)}$ of the quantum cohomology form (a
realisation of) a quantum OPA. We still have to prove that we obtained a quantum version
of the classical \WA. One can check that\col
\be
2y\, \cc\, W^{(1,0)}
\ee
satisfies a Virasoro OPE, where we extracted a factor\col
\be
\cc= {\k\over \k+\tilde{h}}\label{eq:RNccdef}\,,
\ee
which goes to $1$ in the classical limit and $y$ is the index of embedding
defined in \re{eq:RNindexydef}. However, we choose to add the
``diagonal" quadratic combination of the $W^{(0,\alpha_0)}$, which
corresponds to the Sugawara tensor of the affine algebra of the $j=0$
highest weight currents. In other words, we choose $s=1$ in the quantum
analogue of \vgl{eq:RNTsclassdef}. As in the classical case
\cite{classHRFORTW}, this choice for the \emt\ is expected to give that
the other generator s $W^{(j, \alpha_j)}$ are primary. We define\col
\bea
\hat{T}^{\rm EM}&=&
   \frac{\k}{x(\k+\tilde{h})}\bigg( \str \left(\hat{J}_z e_-
   \right) + \str \left( [\t,e_- ] \hat{J}_z\right)\bigg)\nonu
&& +\frac 1 {x(\k+\tilde{h})}
    \str \left(\P_0(\hat{J}_z)\P_0(\hat{J}_z) \right)
 + \frac 1 x\str\left(e_-\dz L\hat{J}_z\right)\nonu
&&       + \frac {1}{x(\k+\tilde{h})}\str\left(
           \left[\P_0 (t^A),\left[\P_0 (t_A)
             ,\dz L\hat{J}_z \right]\right]e_-\right)\nonu
&&  -\frac{\k}{4x}\str \left( [\t,e_-]\dz \t\right)\,,
\label{eq:RNTEMdef}
\eea
where $L$ is defined in \vgl{eq:RNLdef}. The first term is
$2y\cc\left(W^{(1,0)}_0+W^{(1,0)}_1\right)$,
the remainder forms $2y\, \cc\, \left(W^{(1,
0)}_2 + {2\over xy\k} \str (\P_0W)(\P_0W)\right) $.
To bring this in a more recognisable form, we add a BRST exact term
to $\hat{T}^{\rm EM}$\col\footnote{
This formula corrects a misprint in \cite{us1} in the coefficient of
the second term of the second line. }
\bea
\hat{T}^{\rm IMP}
&\equiv&\hat{T}^{\rm EM} +
Q\left(\frac{2}{x(\k+\tilde{h})}\str\, b(J_z+\frac 1 4 \{b,c\})\right)\nonu
&=&\frac{1}{x(\k+\tilde{h})} \str J_zJ_z -
  \frac{1}{2x}\str e_0 \dz  J_z
-\frac{\k}{4 x}\str\left( [\t,e_-]\dz \t\right)\nonu
&&+\frac{1}{4x}\str b[e_0,\dz  c]-\frac {1}{2x}\str b\dz
c+\frac{1}{4x}\str\dz  b [e_0,c]
\,.\label{eq:RNBTimp}
\eea

We now analyse the various parts of the improved \emt,
\vgl{eq:RNBTimp}. The fact that it differs by a BRST exact form from
the $T^{\rm EM}$ does not change the value of the central charge. The
first term is the Sugawara tensor \re{eq:Sugawara} for the \KA\
$\hat{g}$ with central charge\col \be
c_{\rm Sug}=\frac{\k(d_B-d_F)}{\k+\tilde{h}}\,.\label{eq:RNcccS}
\ee
Here $d_B$ and $d_F$ are the number of bosonic and fermionic
generators of $\bar{g}$ .\\
The second term of \vgl{eq:RNBTimp} is the so-called improvement term. Given
an affine current $J\in\P_m\bar{g}$, it changes the
conformal dimension of $J$ from $m$ to $m+1$. It was added by hand in
\cite{classHRFORTW} to ensure that all terms in the constraint
$\str\left( (J_z -\k/2 e_-)e_+\right)$ have conformal dimension zero.
The contribution of the second term to the central charge is\col
\be
c_{\rm imp}=-6y\k\,,
\ee
In the case where $\tilde{h}$ is different from zero, we can
evaluate \vgl{eq:RNindexydef} in the adjoint representation by using
\vgl{eq:Killingdef}\col
\bea
y=\frac{1}{3\tilde{h}}\sum_{j\alpha_j}(-1)^{\alpha_j}
j(j+1)(2j+1)\,, \label{eq:RNBindexem}
\eea
and $(-1)^{\alpha_j}=+1$ ($-1$) if $t_{(jm,\alpha_j)}$ is bosonic
(fermionic).

The next term of \vgl{eq:RNBTimp} is the \emt\ for the auxiliary $\t$
fields with central charge\col
\be
c_\t=\frac 1 2 \left(\dim \left(\P^F_{1/2}\bar{g}\right) -
    \dim \left(\P^B_{1/2}\bar{g}\right)\right)\,,
\ee
where the superscript $B\, (F)$ denotes a projection on the bosonic
(fermionic) part of the algebra.

Finally, the last terms of \vgl{eq:RNBTimp} form the \emt\ for the
ghost--anti\-ghost system $T_{\rm gh}$. To each generator $t$ of the
gauge group, $t\in\P_m\bar{g}$ where $m>0$, we associated a ghost
$c\in\P_m\bar{g}$ and an antighost $b\in\P_{-m}\bar{g}$. They have
respectively conformal dimension $m$ and $1-m$ with respect to $T_{\rm
gh}$. Such a pair contributes $\mp 2(6 m^2-6m+1)$, where we have the minus
sign if $b$ and $c$ are fermionic, and a plus when they are bosonic, see
\vgl{eq:ccbc}. As such, the total contribution to the central charge
coming from the ghosts is given by\col
\be
c_{\rm gh}=-\sum_{j,\alpha_j}(-1)^{\alpha_j}2j(2j^2-1)+\frac{1}{2}\left(
\dim \left(\P^B_{1/2}\bar{g}\right) -\dim
\left(\P^F_{1/2}\bar{g}\right)\right)\,.\label{eq:RNcccg}
\ee
Adding all this together, we obtain the full expression for the total
central charge $c$ as a function of the level $\k$\col
\be
c=c_{\rm Sug}+c_{\rm imp}+c_\t+c_{\rm ghost}\,.\label{eq:RNccct}
\ee
Using the explicit form for the index of embedding $y$ given in
\vgl{eq:RNBindexem} and some elementary combinatorics, we can rewrite
\vgl{eq:RNccct} in the following pretty form\col
\be
c=\frac 1 2 c_{\rm crit} - \frac{(d_B-d_F)\tilde{h}}
{\k+\tilde{h}} - 6 y (\k+\tilde{h}),\label{eq:RNBcpretty}
\ee
where $c_{\rm crit}$ is the critical value of the central charge for the
extension of the Virasoro algebra under consideration\col
\be
c_{\rm crit}=\sum_{j,\alpha_j}(-1)^{\alpha_j}2(6j^2+6j+1)\,.
\label{RNBccrit}
\ee

Knowing the leading term of the generators $W^{(j, \alpha_j)}$,
\vgl{eq:RNBlterm0}, we find that their conformal dimension is $j+1$.
We are convinced that they are primary with respect to $T^{\rm EM}$, but
leave the verification of this to the industrious reader.

We now come to the main result\col\
\begin{theorem}
The generators $W^{(j,\alpha_j)}$
form an extension of the Virasoro algebra with $n_j (\mbox{\small
adjoint})$ generators of conformal dimension $j+1$ and with the value
of the central charge given by \vgl{eq:RNBcpretty}.
\end{theorem}

To finish the proof of this statement, it only needs to be shown that
the fields $W^{(j,\alpha_j)}$ close under OPEs. From the fact that the OPEs
close on $\wha$ and preserve the grading, one deduces that the OPEs of
the generators $W$ close modulo BRST exact terms. However, there
are no fields of negative ghostnumber in the reduced complex $\wha$. This
implies that the OPEs of the generators $W$ close.

\section{An example : $\asp$, continued\label{sct:OSPpart2}}
We now continue our example of section \ref{sct:OSPpart1}
for the quantum case using the results of the
previous section. For this we need the OPEs of $\tau$ and the (anti)ghosts.
They follow from the action $\cs_{\rm gf}$ \re{eq:RNBgauges1}, see intermezzo
\ref{int:RNFFOPEs} and \cite{us1}. We can use the
results for the cohomology of the BRST operator of the previous section.
In the $\asp$ case, $L\hat{J}_z $  is given by $-\hat{J}^0_z t_{\pp}$. We
get the following representants of the cohomology of $Q$\col
\bea
W^{(0,i)}&=&\hat{J}_z^i+
    \frac{\k}{2\sqrt{2}} \l_{ab}{}^i\t^{+a}\t^{+b}\nonu
W^{(1/2,a)}&=&\hat{J}_z^{+a} +\sqrt{2}\l_{ab}{}^i\hat{J}_z^i \t^{+b}+
   \hat{J}_z^0\t^{+a}-\frac {\k+1}{2}\del\t^{+a}\nonu
W_s^{(1,0)}&=& \hat{J}^{\pp}_z
   +\frac 2 \k \hat{J}^0_z \hat{J}^0_z
   +2\hat{J}^{+a}_z  \t^{+a}
   -\sqrt{2}\l_{ab}{}^i\hat{J}_z^i \t^{+a} \t^{+b}
   -\frac{\k+1}{\k}\del \hat{J}^0_z \nonu
   &&\,\, -\frac{2\k+3}{4} \del \t^{+a} \t^{+a}
   +s \frac{2}{\k}W^{(0,i)} W^{(0,i)} \,.
\label{eq:RNospcohogenqua}
\eea
These expressions are the same as the classical ones (eqs.\
\re{eq:RNBpolys} and \re{eq:RNBTsdef} with $J_z\rightarrow\hat{J}_z$), up
to finite renormalisation factors related to normal ordering.
The operator $2\k/(\k +\tilde{h})\, \hat{T}_s^{(1, 0)}$ satisfies the
Virasoro algebra for $s=1$ or $(-2\k + N -3)^{-1}$, which corresponds to
the classical values \re{eq:RNBTsdef} in the large $\k$ limit.

We checked using \OPEdefs\ (with an extension to use dummy arguments),
that these currents (with $s=1$) form a
representation of the quantum $\soN$ superconformal algebra
\re{eq:RNSO(N)algebra} with the normalisation factors\col
\bea
G^a=4i\sqrt{\k\over \k+\tilde{h}} W^{(1/2,a)}&{\rm and}&
U^i=-{2\sqrt{2}}W^{(0,i)}\,,\label{eq:RNospnorms}
\eea
and the value for the central charge as given by \vgl{eq:RNBcpretty}.
Alternatively, we can express the level of the $\soN$ \KA\ in terms of $\k$\col
\be
k=-2\k-1\label{eq:RNsoNkkappa}\,.
\ee

Comparing \vgl{eq:RNospnorms} to \vgl{eq:RNSONgeneratorsclass} we see
that the quantum currents have an overall renormalisation factor $\cc^j$,
where $\cc$ is defined in \vgl{eq:RNccdef}.

\section{Quantum corrections to the extended action\label{sct:RNSBVq}}
Using the results on the cohomology of the BRST operator \re{eq:RNBQdef}
in section \ref{sct:cohomology}, we will now determine the
quantum corrections to the gauge-fixed action \re{eq:RNBactiongen2}.
As already explained, we find the corrections by requiring BRST
invariance of the gauge-fixed action in a point-splitting regularisation
scheme.

No modifications to the terms which are independent of $\mu$ are needed.
The fields $\hat{\ct}^{(j, \alpha_j)}$ proportional
to $\mu_{(j, \alpha_j)}$ in \vgl{eq:RNBactiongen2} have to be elements of
the quantum cohomology. The classical $\hat{\ct}^{(j, \alpha_j)}$ can be
viewed as functionals of the generators of the classical cohomology
$W^{(j, \alpha_j)}_{\mbox{\small classical}}$. To find the quantum
corrections to $\hat{\ct}$, we cannot simply substitute the generators of the
classical cohomology $W$ by their quantum counterparts. Indeed, for the
Virasoro operator we proved in subsection \ref{ssct:RNWA} that an
additional renormalisation factor $\cc$, given in \vgl{eq:RNccdef}, is
necessary to preserve the Virasoro OPE. The example of section
\ref{sct:OSPpart2} suggests that a convenient renormalisation for the
other generators is given by\footnote{ This is a slightly different
choice as used in \cite{us3}.}\col
\be
\hat{W}^{(j,\alpha_j)}\ =\ \cc^j W^{(j,\alpha_j)} \,,
\label{eq:RNWhatnorm}
\ee
where the normalisation of the fields $W^{(j,\alpha_j)}$ is fixed in
\vgl{eq:RNWAgendef}. In summary, the quantum corrections to the
gauge-fixed action \re{eq:RNBactiongen2} in the gauge $A_\zb=0$
are obtained by replacing the classical $W$ currents with the
quantum $\hat{W}$ in $\hat{\ct}^{(j,\alpha_j)}$, which we write
as $\hat{T}[\hat{W}]$.

Because $\hat{\ct}$ does not contain $J^*_z, c^*, \t^*$ or
$A_\zb$, we can extend these results to the extended action.
Also, no modifications to the terms which arise
from the quantum transformation laws are needed, see table
\ref{table:RNBqonf}.
In this way, we have determined the quantum BV
action completely in an OPE regularisation. Giving the antighosts $b$
their original name $A^*_\zb$, we find\col
\bea
\cs^q_{\rm BV} &=& \k S^-[g]+ \frac{1}{\pi x} \int \str\,
     A_\zb\left( J_z  + {1\over 2}\{A^*_\zb,c\}-\frac \k 2 e_- -\frac \k 2
     [\t,e_-]\right)\nonu
 &&+ \frac{\k}{2\pi x}\int \str \left(
    {1\over 2} [\t,e_-]\db\t - c^*cc + \t^*c
     + A^*_\zb\db c \right)\nonu
 && + \frac{\k}{2\pi x}\int \str \left(
     J^*_z\left(\frac \k 2 \del c + [c,J_z] \right)\right)\nonu
 && +{1\over 4\pi xy}\int\str\left(\mu\,\hat{\ct}[\hat{W}]\right)\,.
\label{eq:RNSqbv} \eea
We feel confident that $Q^2\!=\!0$ guarantees the gauge invariance of
the quantum theory. Therefore we will use \vgl{eq:RNSqbv} as it stands,
also for a different gauge in chapter \ref{chp:renormalisations}.


\section{Discussion}
We have proven that any (super) Lie algebra with an $\slt$ embedding gives
rise to a realisation of a (classical and quantum) \WA,
which is generated by $n_j (\mbox{\small
adjoint})$ generators of conformal dimension $j+1$. \WA s with
dimension $1/2$ fields cannot be realised by a Drinfeld-Sokolov reduction.
However, this is no shortcoming of the present approach since such fields
can always be factored out, as we have explained in chapter
\ref{chp:FactFree}.

For a classical bosonic \WA, it was shown in 
\cite{Bowcockcontraction,Dublin:vpa} 
that one can identify a finite Lie algebra with an embedded
$\slt$ in the mode algebra, called the ``vacuum preserving algebra'',
see also subsection \ref{ssct:BowWatts}.
Furthermore, for a \WA\ constructed by a Drinfeld-Sokolov reduction of a
Lie algebra $\bar{g}$, this Lie algebra is isomorphic to $\bar{g}$. We
expect that the results of section \ref{sct:cohomology} on the quantum
Drinfeld-Sokolov reduction can be used to prove that the vacuum
preserving algebra of the quantum \WA\ and the classical \WA\ are the
same.

For any quantum \WA\ arising in a Drinfeld-Sokolov
reduction, we have constructed a path integral formulation of the induced
action in terms of a gauged \WZW. In a point-splitting regularisation, the
results on the quantum cohomology give the quantum corrections to the gauge
fixed action to all orders. This will be used in the next chapter to
discuss the effective action of the corresponding induced $\Ww$--gravity
theory.



