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\newsec{Further developments}

\subsec{$\cW$-gravity}

In this section we will briefly review some developments in
(classical and quantum) $\cW$-gravity.
$\cW$-gravity is a higher-spin extension of $d=2$ gravity whose
structure is based on an underlying $\cW$-algebra. This $\cW$-algebra
takes over the role played by the Virasoro algebra in pure $d=2$ gravity.
The main applications of $\cW$-gravity, which are in the area of
string theory, will be discussed in the next section. Review papers
on $\cW$-gravity are for example \cite{\BBS,\Huc,\SSvNf}.

\medskip

It is well-known that classical theories of gravity or supergravity
(in general dimensions) can be constructed in a systematic way
by starting from an algebra of space-time (super)symmetries. This
is done by first constructing a gauge theory for the symmetry algebra
and then imposing a number of Yang-Mills curvature constraints.
Because of these constraints general coordinate transformations become
a local symmetry, and in this way a theory of (super)gravity arises
\cite{\KTN}.
For $d=2$ $\cW$-gravity this procedure was successfully applied
in \cite{\SSvNc}, where a covariant Lagrange formulation of classical
$\cW_3$ gravity coupled to scalar fields was presented.

The gauge algebra used in \cite{\SSvNc} for the construction of covariant
$\cW_3$ gravity is a centerless classical limit of the quantum $\cW_3$
algebra. After solving the Yang-Mills curvature constraints in the
corresponding gauge theory, one finds a gauge multiplet which
contains four
zweibein fields $e_\mu{}^+$, $e_\mu{}^-$ and four $\cW_3$ zweibein fields
$b_\mu{}^{++}$, $b_\mu{}^{--}$, where $\mu=\pm$. There are
eight local gauge symmetries: general coordinate, Weyl and local
Lorentz symmetries, together
with their $\cW_3$ analogues. The coupling of these
$\cW_3$ gravity fields
to scalar matter fields was worked out in \cite{\SSvNc}, where a gauge
invariant kinetic action for $N$ scalar fields $\phi^i$, $i=1,2,\ldots,N$
was presented. We shall not display this action here, but instead discuss
the results in the chiral gauge and in the light-cone gauge, which can
both be obtained by fixing some of the local symmetries.

In the chiral gauge
the coupling of scalar matter fields to $\cW_3$ gravity, which was
first obtained in \cite{\Hua}, is easily
explained. We start from the observation \cite{\Hua} that a free
action for $N$ scalar fields admits a rigid $\cW_3$ symmetry.
The transformations of the scalar fields read
\eqn\chHaa{
\delta_\epsilon \phi^i = \epsilon_+ \, \del_- \phi^i \,, \quad
\delta_\lambda \phi^i = \lambda_{++} \, d^{ijk} \,
\del_- \phi^j \del_- \phi^k \,,
}
where $d^{ijk}$ is a symmetric 3-index tensor. [We write $\del_-$ and
$\del_+$ for $\del_z$ and $\del_{\zb}$, respectively.] One can promote
these rigid
symmetries to local gauge invariances by introducing gauge fields
$h_{++}$ and $b_{+++}$ in the standard way. It turns out \cite{\Hua} that
the scalar field action with only the minimal coupling to these gauge
fields,
\eqn\chHab{
S_{\rm ch} = \frac{1}{\pi} \int d^2 z \; \left[
    - \half \del_+ \phi^i \del_- \phi^i + \half h_{++} \del_+ \phi^i
    \del_+ \phi^i + \frac{1}{3} b_{+++} d^{ijk}
    \del_+ \phi^i \del_+ \phi^j \del_+ \phi^k \right] \,,
}
is gauge-invariant, provided we choose the transformation rules of
$h_{++}$ and $b_{+++}$ appropriately and we have the identity
\eqn\chHac{
d^{k(ij} \; d^{l)mk} = \delta^{(ij} \; \delta^{l)m} \,.
}
The two local symmetries, with parameters $\epsilon_+(z,\zb)$ and
$\lambda_{++}(z,\zb)$, can be viewed as particular linear combinations of
all eight local symmetries present in the covariant formulation.

In the light-cone gauge, with both chiralities present, the coupling of
scalar matter fields to $\cW_3$ gravity is more involved.
The $\cW_3$ gauge
fields are $h_{\pm\pm}$ and $b_{\pm\pm\pm}$, corresponding to local
symmetries with parameters
$\ep_\pm(z,\zb)$ and $\la_{\pm\pm}(z,\zb)$. In \cite{\SSvNb} it
was found that the
light-cone gauge action is non-polynomial in the spin-3 gauge fields.
This
difficulty can be circumvented by introducing auxiliary fields $F_+{}^i$
and
$F_-{}^i$, which will play the role of so-called {\it nested covariant
derivatives}. With these variables, the gauge-invariant action takes the
following form \cite{\SSvNb}
\eqn\chHad{
\eqalign{
S_{\rm lc} = & \frac{-1}{\pi} \int d^2 z \, e \left[
  - \half \nabla_+ \phi^i \,\nabla_- \phi^i - F_+{}^i F_-{}^i +
  F_+{}^i \left( \nabla_- \phi^i - \frac{1}{3} b_{---}
  d^{ijk} F_+{}^j F_+{}^k \right) \right.
\cr
&  \quad \quad \quad \left. + F_-{}^i \left( \nabla_+ \phi^i -
   \frac{1}{3} b_{---} d^{ijk} F_-{}^j F_-{}^k \right) \right] ,
\cr}
}
where $e=(1-h_{++}h_{--})^{-1}$ and $\nabla_{\pm} = \del_{\pm} -
h_{\pm\pm}\del_{\mp}$. The field equation of $F_-{}^i$ is algebraic and
leads to
\eqn\chHae{
F_+{}^i = \nabla_+ \phi^i - b_{+++} d^{ijk} F_-{}^j F_-{}^k
}
(with a similar result for $F_-{}^i$).  When solving $F_\pm^i$ by
iteration one obtains a $\cW_3$ generalization of a covariant derivative,
which is infinitely nonlinear and is appropriately called a nested
covariant derivative.

In a similar way, light-cone gauge and covariant actions for $w_\infty$
gravity coupled to scalar fields were constructed in \cite{\BPRSSS}
and in \cite{\SSvNca}, respectively. A geometrical interpretation of
the full non-linear structure of $w_\infty$ gravity was presented
in \cite{\Huba}. We refer to \cite{\BBS,\Hub,\Mik} for further results on
classical $\cW$-gravity.

\medskip

At the classical level, $\cW$-gravity does not have
degrees of freedom, since all the field components are compensated
for by local gauge symmetries (up to possible moduli). However, upon
quantization one or more of these symmetries become anomalous and
therefore quantum $\cW$-gravity does have degrees of freedom.
If the central charge $c$ of the matter system coupled to
$\cW$-gravity is tuned to a specific value, the quantum anomalies
cancel and the $\cW$-gravity degrees of freedom decouple. This is
similar to what happens in $d=2$ induced gauge theories and
$d=2$ ordinary gravity \cite{\Pob,\SSvNf}.

The dynamics of the quantum degrees of freedom of
$\cW$-gravity are governed by an {\it effective} action.
This action can be studied for various gauge
choices, for example in the chiral gauge or in the conformal gauge.
In the chiral gauge the effective action arises in the following way
\cite{\Mat,\SSvNd,\Hubb,\SSvNe,\GvN}.

One first obtains an {\it induced} action by performing a functional
integral over the matter fields, of total central charge $c$, to which
the $\cW$-gravity fields are coupled.
The non-trivial contributions to the induced action arise from
loop-diagrams with $\cW$-gravity fields on external lines and
matter fields
in the loops. The resulting induced action, which can be viewed as an
integrated anomaly, is non-local. The {\it effective} action is then
obtained by renormalizing this induced action, taking into account
loop-diagrams for the anomalously propagating $\cW$-gravity fields.
The entire computation of the effective action can be done
perturbatively, with $1/c$ as the expansion parameter.

In \cite{\dBGa,\dBGb}
the covariant induced action for $\cW_n$ gravity (in the limit
$c\to\pm\infty$) was presented. It is given in terms of
two (left and right) chiral sectors, which contain fields
$b_\pm^{(i)}$, $i=2,\ldots,n$, $(n-1)$ scalar fields $\ph^{i}$,
$i=2,\ldots,n$, and a number of auxiliary fields which play
the role of nested covariant derivatives as in \chHad, \chHae.
When specializing to the conformal gauge, one finds
\cite{\dBGa} that
this induced action takes the form of a Toda action for the scalar fields
$\ph^i$. This implies that in the conformal
gauge the $\cW$-currents of $\cW$-gravity take the familiar
free-field form (compare with \chFcc, \chFce).
Because of this, explicit computations in quantum $\cW$-gravity, such as
for example the analysis of the BRST cohomology of physical states, are
most easily done in the conformal gauge \cite{\DDR}.

\medskip

In the above we mentioned that the effective action for
$\cW$-gravity can be computed perturbatively in quantum field theory,
with $1/c$ as the expansion parameter. However, it is possible to
exploit once more the relation between $\cW$-algebras and affine
Lie algebras to obtain results exact to all orders for some
of the quantities in $\cW$-gravity! The idea here is that
$\cW$-gravity can be obtained by constraining WZW field
theory or, equivalently, by reducing $d=3$ Chern-Simons gauge
theory. We will now briefly explain how this works for
$\cW_3$ gravity.

Consider $\cW_3$ gravity coupled to
a $\cW_3$ CFT of central charge $c$.
The matter sector can be described in the QDS
reduction scheme (see Chapter 6), where the starting point is the affine
Lie algebra $\widehat{sl}(3)$ at level $k$, with (see \chFbg)
\eqn\chHa{
c= c_k = 2- 24 \left( \sqrt{1\over k+3} - \sqrt{k+3} \right) ^2
= 50 - 24 \left( {1 \over k+3} +k+3 \right) \ .
}
To couple this system to $\cW_3$ gravity, we
must make sure that the central charge of the $\cW_3$
gravity sector compensates the matter central charge $c_k$ {\it plus} the
contribution of the $\cW_3$ ghosts, which equals $c_{gh}=-100$
(see Section 8.2). Observe that $c_k = 100 - c_{-(k+6)}$,
which shows that we obtain the correct contribution to the central
charge if we apply the QDS reduction to an $sl(3)$ WZW model of level
$\kappa = -(k+6)$. With \chHa\ this gives
\eqn\chHb{
\kappa = - {1 \over 48} \left( 50-c + \sqrt{(c-98)(c-2)} \right) -3 \ ,
}
where the sign in front of the square root has been chosen in
accordance with the classical limit $c \rightarrow -\infty$, in which
$\kappa \sim c/24$.

The picture that arises at this point is the following: we can represent
the degrees of freedom of the {\it effective} quantum  $\cW_3$
gravity, induced  from a matter system of central charge $c$
and after renormalization,
by a QDS reduced $sl(3)$ WZW model of level $\kappa$
as in \chHb. This observation explains the presence of a `hidden'
affine $sl(3)$
symmetry in $\cW_3$ gravity, which generalizes the affine $sl(2)$
symmetry in ordinary gravity first observed by Polyakov \cite{\Poa}.
For ordinary gravity the connection with a constrained $sl(2)$ WZW model was
first understood in \cite{\KPZ}; the generalization to $sl(n)$ was worked
out in \cite{\BOa}.

{}From the above picture one expects that the effective action for
$\cW_n$-gravity can be obtained in explicit form by reducing
the $sl(n)$ WZW action.
For $\cW_3$ gravity in the chiral gauge this result has been checked
by explicit perturbative computations.
It was found \cite{\Mat,\BFK,\DHR,\OSSvN} that the induced
action for chiral $\cW_3$ gravity, in the limit $c \rightarrow
\pm\infty$, is governed by local Ward identities which can be
obtained by reducing the Ward identities for the $sl(3)$ WZW model.
For finite $c$, the induced action contains additional
terms that correspond to non-local additional terms in the Ward
identities \cite{\SSvNd}. However, the results of \cite{\SSvNe}
suggest that these extra terms get cancelled if one renormalizes the
induced action. In \cite{\SSvNe} the full effective action was
computed through the first non-leading order in the perturbative
$1/c$ expansion,
and it was found that the result precisely takes the form of a
reduced WZW action with renormalized coefficients. The overall
coefficient of the effective action can be identified with the
level $\kappa$ of the affine $sl(3)$ algebra. The perturbative result
for $\kappa$ thus obtained
agrees with the relation \chHb\ through the first
non-leading order in $1/c$.

In \cite{\OSSvN,\SSvNf} the effective action for chiral
$\cW_3$ gravity was given in a closed form.
In terms of Polyakov type variables $f$ and $g$,
this action is {\it local}, and it generalizes the effective
action for ordinary gravity in terms of Polyakov's variable $f$ \cite{\Poa}.

\subsec{$\cW$-symmetry in string theory}

We would now like to come back to some
remarks we made in Chapter 1, namely the applications
of $\cW$-symmetry in string theory. Many of the important
issues in this field
are still unsettled and the discussion below is intended to merely
give a flavor of the developments that are taking place.

In the standard formulation of string theory, the fields representing
the coordinates of a (first-quantized) string in target space-time,
define a `matter' CFT.
The reparametrization invariance on the
string world-sheet implies that this CFT should be coupled
to two-dimensional (world-sheet) gravity. If the total
central charge of the `matter' CFT
is equal to 25,  the $d=2$ gravity sector decouples.
In that case the gravity sector adds one free scalar
field to the space-time coordinates, and it leads to a number of
constraints (the Virasoro constraints) on physical states in the
string theory. If the
matter central charge differs from 25 the extra scalar field becomes
interacting and the situation is more intricate.

In recent years, considerable progress has been made in understanding
the coupling of $c\leq 1$ (minimal) CFT's to $d=2$ gravity.
It has turned out that these theories
can also be studied from the point of view of discretized world-sheets
(matrix models) or from the point of view of topological field theories.
For $c>1$, which is the regime where the bosonic string develops
tachyonic states, the coupling to $d=2$ gravity has run into
`strong-coupling problems' \cite{\KPZ} and has not yet been understood
properly.

\medskip

In the study of $c<1$ CFT's coupled to gravity, interesting
connections with $\cW$-symmetry have come up.
Namely, it was
found, both in the matrix model formulation and in the topological
approach, that the partition function of the theory (as a function of
a set of coupling constants) can be characterized by so-called
$\cW$-constraints \cite{\DVVb,\FKNa,\Goe}. The appearance of these
constraints is closely related to the fact that these models can be
analyzed in terms of generalized KdV hierarchies \cite{\Doc}.
The paper \cite{\FKNb} shows that the $\cW_n$ constraints can
be viewed as reductions of more general $\cW_{1+\infty}$ constraints,
which arise naturally when one views the generalized KdV hierarchies
as reductions of the KP hierarchy.

The study of $c=1$ strings, both in terms of matrix models
and as a continuum theory, has revealed an interesting symmetry
structure. In the case of a flat, uncompactified background the
(chiral) symmetries in the continuum theory have been identified
\cite{\KPa,\Wid} with the area preserving polynomial vector-fields,
generating (the wedge of)
a $w_{\infty}$ algebra as in \chEac.

\medskip

In the cases just mentioned $\cW$-symmetries arise as `bonus'
symmetries in specific string theories that were constructed
without any reference to $\cW$-symmetry. It is certainly
interesting to see if one can construct string theories
with manifest $\cW$-symmetries built in.
A first step in this direction is to study $\cW$-extensions
of $d=2$ gravity and their couplings to CFT. In Section 8.1 we
briefly reviewed some results in this area. We will now
further explore the possibilities to construct $\cW$-extensions
of string theories.

A first remark concerns the $\cW$-constraints in matrix models
and topological field theories mentioned above. It has been proposed
\cite{\AQ,\DDR} that the coupled systems of CFT plus gravity, for which
these constraints occur, are actually closely related to theories
of {\it pure} $\cW$-gravity. Physical states in the spectrum of pure
$\cW$-gravity can sometimes be viewed as CFT matter states `dressed'
by the fields of pure $d=2$ gravity \cite{\DDR}. However, full equivalence
of both theories cannot be claimed, and the connection remains rather
mysterious.

Going one step further, one can consider $\cW$-invariant CFT's coupled
to the corresponding $\cW$-gravity. One interesting observation
is that in this context the critical central charge $c=1$ for
ordinary gravity gets shifted to a higher value. For example, 
for the $\cW$-algebras
related to the $ADE$ simply-laced Lie algebras the threshold value
for the central charge would be the rank $\ell$ of the Lie algebra.
[For $\cW_3$, where $\ell=2$, this can be read off from equation
\chHb; other cases can be treated similarly.]
This makes clear that certain CFT's with $c>1$, whose coupling to
ordinary gravity is problematic, can consistently be coupled to
$\cW$-gravity. We expect that such can be done for {\it any} RCFT,
where the $\cW$-algebra for the $\cW$-gravity theory should be closely
related to the chiral algebra of the RCFT.

Systems of $\cW$-invariant CFT coupled to $\cW$-gravity can tentatively
be interpreted as (critical or non-critical) $\cW$-strings. For the case
of $c<\ell$ minimal models of one of the $ADE$ $\cW$-algebras coupled to
the corresponding theory of $\cW$-gravity, one expects behavior
which is qualitatively similar to that of the standard $c<1$ strings.
In particular, one expects connections with generalized matrix models
and topological field theories, which still remain to be worked out.

The direct generalization of lower critical ($c_1=1$) and upper critical
($c_2=26$) bosonic strings to $\cW$-strings is problematic.
On the level of the algebra the numerology is clear: for example, for the
$ADE$ $\cW$-algebras the lower critical dimension is $c_1=\ell$, and
the upper critical dimension has been found to be
$c_2=2\ell (2{\rm h}^2+2{\rm h}+1)$
\cite{\BBSSa}, where ${\rm h}$ is the Coxeter number
of the Lie algebra. The latter
value is the one for which a nilpotent BRST charge is expected to exist.
For a general (generic) $\cW$-algebra, the critical central charge is
given by
\eqn\chHba{
c = \sum_s \, 2\, (-1)^{2s}\, (6 s^2 - 6 s +1) \ ,
}
where the summation runs over the spins $s$ of the independent generators
of the $\cW$-algebra. However, the existence of a
nilpotent BRST charge is not always guaranteed
(due to complications with non-linearity, see \cite{\SSvNa}) and
should be checked in each individual case.
For the $\cW_3$ algebra, where $c_2=100$, a nilpotent quantum
BRST charge has been constructed in \cite{\Tha,\SSvNa}.
[For infinite $\cW$-algebras the sum \chHba\ needs to be
regularized. It was argued in \cite{\Yab} that for the
$\cW_\infty$ algebra a nilpotent BRST charge exists for $c=-2$.]

The facts
that the notion of critical central charges $c_1=1$ and $c_2=26$ can be
generalized to (at least) the $ADE$ $\cW$-algebras, and that nilpotent
BRST charges can presumably
be constructed, do not by themselves
imply the existence of interesting new string theories. In addition,
a string theory requires that (part of)
the matter conformal field theory
can be viewed as a set of coordinates on a target space-time. In practice
one should therefore consider realizations of (critical or non-critical)
$\cW$-algebras in terms of scalar matter fields. For the $\cW_3$ algebra
realizations in terms of an arbitrary number of scalar fields
(and with adjustable central charge) were discussed in \cite{\Roa}.
The anomaly-free coupling of such matter systems with central
charge $c=100$ to $\cW_3$ gravity was discussed in \cite{\PRSta}.

In general, scalar field realizations of $\cW$-algebras involve
background charges, which lead to mass-shifts in the spectrum of physical
string states. Because of that, the original expectation
that the spectrum of a $\cW$-string might contain states with space-time
spin greater than two \cite{\BGe} is probably not justified.
Also, the fact that the equations determining the BRST cohomology of
physical states involve higher ($\geq 2$) order polynomials in general,
leads to a `branching' of the spectrum of physical states.
As a consequence, in the lower critical dimension $d=\ell$,
$\cW$-strings for the $ADE$ $\cW$-algebras are {\it not}\ free
from tachyons \cite{\DDR}.
In fact, it was argued in \cite{\DDR} that the critical numbers $d_1$ and
$d_2$ of scalar fields in $\cW_n$ gravity (as opposed to the critical
central charges) are $d_1=6/n(n+1)$ and $d_2 = 24 + 6/n(n+1)$.
We would like to stress, however, that the validity of these results
depends crucially on a specific (but
debatable) Ansatz for the matter couplings in the conformal gauge.

The spectrum of critical $\cW_n$ strings has been further analyzed in
\cite{\LPSX}. In \cite{\LPScW} some of these results were generalized
to $\cW$-strings based on more general $\cW$-algebras.

\medskip

String theory is more geometrical than CFT, which seems to be
one of the reasons why the application of finitely generated
$\cW$-algebras to string theory is rather intricate. There have been
a number of proposals for a more geometrical understanding of $\cW$-symmetry,
but their possible applications to string theory have not been clarified.

Some related ideas concerning the geometrical structure of
$\cW$-symmetries have been proposed in the papers \cite{\SSZ,\SoS,\GM},
see also \cite{\Biba,\GLM,\BFK}. The paper \cite{\GM} associates
$\cW$-symmetries
with the extrinsic geometry of the embedding of two-dimensional manifolds
with chiral parametrization into higher dimensional K\"ahler manifolds.
The characteristic equations for such embeddings can be connected to a
Lax pair for certain Toda equations, and these then form the link to
$\cW$-symmetries.

The constructions of $\cW$-gravity based on
Drinfeld-Sokolov reduction and on the connection with $d=3$ Chern-Simons
theory (see Section 8.1) suggest an alternative way to understand the
geometry of $\cW$-symmetries. These ideas have been essential for the
explicit construction of the covariant induced action of
$\cW_n$ gravity \cite{\dBGb}.

\medskip

In conclusion, we would like to stress that the status of $\cW$-symmetry
in the context of CFT is much better understood than the role these
symmetries can play in string theory.
The precise interpretation, in the context of string theory,
of results obtained for $\cW$-gravity and $\cW$-geometry
is not always clear. However, the fact that already `toy models'
of $c<1$ and $c=1$ strings exhibit $\cW$-symmetries certainly suggests
that extended symmetries will eventually also play a role in more
realistic theories of first and second quantized strings.

\vskip 4mm

\noindent {\it Acknowledgements.}\ We would like to thank the editors
of Physics Reports for the invitation to write this review. It is a
pleasure to thank Krzysztof Pilch, Jos\'e Figueroa-O'Farrill,
Alexander Sevrin, Tjark Tjin and Changhyun Ahn for
carefully reading parts of the manuscript and suggesting improvements.
In particular we would like to thank K.~P.
for pointing out our `crimes against the English language', and
J.~F-O'F. for sharing his \TeX nological insights.
P.B. would like to thank the I.T.P. at Stony Brook for their
hospitality and financial support during the course of this work.
Finally,
we would like to express our gratitude to many of our colleagues at
CERN and Stony Brook for their ever continuing encouragement to
complete this paper.
\bs

\noindent {\it Note added.}\ 
An interesting problem that has not been solved is to
generalize the explicit expression for the coset spin-3 
generator, as discussed in section 7.3.2, to the other
coset generators of spin greater than 3. In the preprint
version of this paper we gave some expressions for these
which are however incorrect. The problem can be traced back 
to a higher-spin generalization of (7.26) that was first proposed 
in \cite{\Thb} (equation (2.12)). Inconsistencies in the coset 
expressions derived from this equation show that it cannot be 
correct (see also the erratum to \cite{\Waa}). It is now
understood that the equation fails to hold because it assumes
a tensor identity for higher order $d$-symbols which is simply
not valid. We would like to thank G. Watts for pointing out this 
problem and A. Sudbery for correspondence on the tensor 
identities.

\vfil\eject

%--------------------------------------------------------------------

\newsec{Appendices}

\noindent{\it A. Lie algebra conventions}\hfil\bigskip

\def\Co{{\rm h}}
\def\bfg{{\bf g}}

Throughout the report we use the following conventions (see
\eg [\Kad] for more details):\bs

\settabs 6\columns
\+ $\bfg$ & a finite-dimensional complex semi-simple Lie
(super)algebra\cr
\+ $\bfh$ & the Cartan subalgebra (CSA) of $\bfg$ \cr
\+ $\bfh^*$ & the dual CSA; we will identify $\bfh$ with $\bfh^*$\cr
\+ $\bfg \cong {\bf n}_-\oplus \bfh \oplus {\bf n}_+$ &
\ \ \ \ a Cartan (triangular) decomposition of $\bfg$ \cr
\+ $(\ , \ )$ &  bilinear form on $\bfh^*$ (sometimes also
denoted by a simple dot $\ \cdot$\ ), \cr
\+ & normalized such that $(\al,\al)=2$ for a long root of $\bfg$ \cr
\+ $\ell$ & the rank of $\bfg$ \cr
\+ ${\rm dim\ }\bfg$ & the dimension of $\bfg$ \cr
\+ $\De_+$ & set of positive roots $\al$ of $\bfg$ \cr
\+ $\al_i$ & a simple root of $\bfg$ \cr
\+ $\La_i$ & a fundamental weight of $\bfg$ \cr
\+ $\rh$ & the Weyl vector of $\bfg$ \cr
\+ $\Co$ & the Coxeter number of $\bfg$ \cr
\+ $Q$ & the long root lattice of $\bfg$ \cr
\+ $P_+$ & the set of integral dominant weights $\la\in\bfh^*$\cr
\+ $W$ & the Weyl group of $\bfg$ \cr
\+ $l(w)$ & the length of the Weyl group element $w$\cr
\+ $\ep(w)$ & the determinant (\ie $\pm1$) of the Weyl group element $w$
\cr
\+ $r_\al\ (r_i)$ & reflection in the root $\al\in \De_+$ (/simple root
$\al_i$) \cr
\+ ${\cal U}(\bfg)$ & the universal enveloping algebra of $\bfg$\cr
\+ ${\cal Z}({\cal U}(\bfg))$ & the center of ${\cal U}(\bfg)$\cr
\+ $h^i$ & a basis of the CSA $\bfh$\cr
\+ $e^\al$ & Lie algebra element corresponding to root $\al$\cr
\+ $a_{ij}$ & the Cartan matrix of $\bfg$, \ie $a_{ij} = 2(\al_i,\al_j)
/(\al_i,\al_i)$ \cr
\+ $e_i\,,i=1,\ldots,\ell$ & the exponents of $\bfg$ \cr
\medskip

The dual Lie algebra $\bfg^*$ is the algebra obtained by inverting
the arrows in the Dynkin diagram corresponding to $\bfg$. Its
corresponding characteristics are denoted by a subscript $\vee$, \ie
$\rhv,\,\dCo,\,P_+^\vee$ etc. In particular the roots $\al^\vee$ of
$\bfg^*$ are related to those of $\bfg$ by $\al^\vee = 2\al/(\al,\al)$.
\smallskip

The untwisted affine Lie algebra $\bfg\otimes \CC[t,t^{-1}]
\oplus \CC c$ [\Kad]
corresponding to $\bfg$ will be denoted by either $\whg$ or
$\bfg^{(1)}$. Characteristics of the affine Lie algebra $\whg$ and
the underlying finite-dimensional Lie algebra $\bfg$ will be
distinguished by putting hats on the former. In addition we will
identify affine weights $\widehat{\La}$ with their finite-dimensional
projection $\La$ supplied by the level $k$. We
use the notation $P_+^{(k)}$ for the set of integral dominant weights
of level $k$. The twisted length $\tilde{l}(w)$ of an affine Weyl
group element $w\in \widehat{W}$ is defined in \eg \cite{\BMPb}.
\bs



We have collected some of the characteristics of finite-dimensional
simple Lie algebras $\bfg$ in Table 1.\bs

\vbox{\offinterlineskip
\hrule
\halign{& \vrule# & \strut\quad\hfil#\quad\cr
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&$\bfg$\hfil&&dim$\,\bfg$ && $h$ && $\dCo$ && $\{e_i\}$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&\cr
& $A_n$ && $n(n+2)$ && $n+1$ && $n+1$ && $1,2,\ldots,n$ &\cr
& $B_n$ && $n(2n+1)$ && $2n$ && $2n-1$&& $1,3,5,\ldots,2n-1$ &\cr
& $C_n$ && $n(2n+1)$ && $2n$ && $n+1$ && $1,3,5,\ldots,2n-1$ &\cr
& $D_n$ && $n(2n-1)$ && $2(n-1)$ && $2(n-1)$ && $1,3,5,\ldots,2n-3,
n-1$ &\cr
& $E_6$ && $78$ && $12$ && $12$ && $1,4,5,7,8,11$  &\cr
& $E_7$ && $133$ && $18$ && $18$ && $1,5,7,9,11,13,17$ &\cr
& $E_8$ && $248$ && $30$ && $30$ && $1,7,11,13,17,19,23,29$ &\cr
& $F_4$ && $52$ && $12$ && $9$ && $1,5,7,11$ &\cr
& $G_2$ && $14$ && $6$ && $4$ && $1,5$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&\cr}
\hrule}
\centerline{\sl Table 1}\bs


\noindent Finally,
we collect here some useful formulae [\Kad]
\eqn\chIaba{ \eqalign{
\rh = \half \sum_{\al\in\De_+} \al\,, & \qquad\qquad
(\rh,\al_i^\vee)=1 \,,\cr
\rh^\vee = \half\sum_{\al\in\De_+} \al^\vee\,,  & \qquad\qquad
(\rh^\vee,\al_i)=1\,,\cr}
}
\eqn\chIac{
{\rm dim\,}\bfg = \ell\,(1+\Co) \,.
}
\noindent The denominator formula:
\eqn\chIaa{
\prod_{\al\in\De_+} (1-e^{-\al}) = \sum_{w\in W} \ep(w) e^{w\rh -\rh} \,.
}
The Freudenthal-de Vries strange formula:
\eqn\chIab{
{ |\rh|^2 \over2\dCo } = { {\rm dim\,}\bfg \over24} \,.
}
And some relations for the exponents:
\eqn\chIad{
\eqalign{
\sum e_i & = \half \ell\,\Co \,,\cr
\sum e_i(e_i+1) & = 4(\rh,\rh^\vee)\,, \cr
& = \third\ell\, \Co(\Co+1) \qquad\qquad {\rm for\ }
\bfg\ {\rm simply\ laced}\,.\cr}
}
\bs\bs


\noindent{\it B. $\cW$-algebra nomenclature}\hfil\bigskip

In this appendix we propose some systematics for naming $\cW$-algebras.
Obviously, this proposal comes somewhat {\it apr\`es la date}, but we
think that it is worthwhile to try and standardize these matters.
We base our notations for $\cW$-algebras on the different approaches to
their construction, which we discuss in the Chapters 5, 6 and 7,
respectively.

\vskip 3mm \noindent
1. We introduce the notion of a $\cW$-algebra of {\it type}
$\cW(2,s_2,s_3, \ldots, s_n)$.
This refers to an algebra that is generated
(in the sense of our definition in Section 3.1) by the
Virasoro generator
$T(z)$ (which is quasi-primary of spin 2) and additional primary
currents of spins $s_2, s_3, \ldots, s_n$. For a $N$-extended
$\cW$-superalgebra
(which contains the $O(N)$-extended superconformal algebra,
$N=1,2,3$ or $4$), we write $\sW^{(N)}(2-{N \over 2},s_2,\ldots,s_n)$.
The first entry refers to the super-Virasoro generator
(which is quasi-primary
of spin $2-{N \over 2}$) and the additional entries $s_2,s_3,\ldots,s_n$
refer to additional generating currents, which are all superfields
in $N$-extended chiral superspace \cite{\Sca}.

Clearly, the type of a $\cW$-algebra does not always completely
fix the algebra. There can be free parameters (the central charge or
still others) and there is a possibility of entirely distinct
algebras with the same set of spins of the generating currents.

\vskip 3mm \noindent
2. In the most general version of the Drinfeld-Sokolov scheme
(see Section 6.3), a $\cW$-algebra is completely determined by a
triple $(\whg,\whg',\ch)$, consisting of an affine Lie algebra
$\whg$, an affine subalgebra $\whg' \subset \whg$ and a 1-dimensional
representation $\ch$ of $\whg'$. We denote the corresponding
$\cW$-algebra by $\cW_{DS}[\whg,\whg',\ch]$. In many cases one
makes a special choice for the defining triple (namely
$\whg'=\widehat{\bf n}_+$, $\ch=\ch_{DS}$) which is completely
determined by the embedding of an $sl(2)$ subalgebra in $\bfg$.
The $\cW$-algebra corresponding to that situation will be called
$\cW_{DS}[\whg,k,\delta]$, where $k$ is the level of $\whg$ and
the vector $\delta$ specifies how the $sl(2)$ subalgebra is embedded
in $\bfg$.

\vskip 3mm \noindent
3. In the coset construction, discussed in Chapter 7, a $\cW$-algebra is
specified by a coset pair $\whg' \subset \whg$, where $\whg$ is an affine
Lie algebra of level $k$ and $\whg'$ is an affine sub-algebra, or by
a pair $\bfg \subset \whg$, where $\bfg$ is the finite dimensional
horizontal subalgebra of $\whg$. (In the latter case the coset
construction
reduces to what we called the Casimir construction or extended Sugawara
construction.) The $\cW$-algebras for such coset pairs will be denoted
by $\cW_c[\whg/\whg',k,*]$, where the $*$ specifies the embedding
$\whg' \subset \whg$, and by $\cW_c[\whg/\bfg,k]$, respectively.

\vskip 6mm

In many cases these notations can be simplified. The subscripts $DS$
and $c$ can be dropped if the context is clear. The embedding data
$\delta$ and $*$ can be defaulted for `obvious' choices, such as the
principal $sl(2)$ embedding for $\delta$, or, for $*$, the diagonal
embedding $\whg \subset \whg \oplus \whg$.

Furthermore, there are `nicknames' for the most familiar algebras.
In particular, there are the $\cW_n$ algebras, of type
$\cW(2,3,\ldots,n)$, which can be realized as
$\cW_{DS}[A_{n-1}^{(1)},k]$,
as $\cW_c[A_{n-1}^{(1)}/A_{n-1},1]$ for $c=n$, or as
$\cW_c[A_{n-1}^{(1)}\oplus A_{n-1}^{(1)}/A_{n-1}^{(1)},(1,k)]$ for
the central charges in the minimal series.
We used the name super-$\cW_3$ algebra for the algebra of type
$\sW^{(1)}(3/2,5/2)$ discussed in Section 3.3.
Similarly, there are the $N=2$ super-$\cW_n$ algebras,
of type $\sW^{(2)}(1,2, \ldots, n)$, which can be realized
as $\cW_{DS}[A(n,n-1)^{(1)},k]$ (see Section 6.3.3).

Algebras that are obtained by applying the DS scheme to various
embeddings of $sl(2)$ in $A_{n-1}$ are sometimes called
$\cW_n^{(l)}$ algebras, where the superscript indicates the embedding
that is used (with $l=1$ denoting the principal embedding).

In the literature the notation $\cW X_\ell$ algebra is often used for
a $\cW$-algebra that one can associate with the Lie algebra $X_\ell$.
This notation works fine for the simply-laced Lie algebras
$X_\ell=A_\ell,D_\ell$
or $E_\ell$ but it is confusing in other cases and we tried as much as
possible to avoid it. We can explain our concern with the example
$X_\ell=B_\ell$.
DS reduction of the affine algebra $B_\ell^{(1)}$ leads to the
$\cW$-algebra $\cW_{DS}[B_\ell^{(1)},k]$, which is of type
$\cW(2,4,6,\ldots,2\ell)$. On the other hand, the coset $\cW$-algebras
$\cW_c[B_\ell^{(1)}/B_\ell,1]$ and
$\cW_c[B_\ell^{(1)} \oplus B_\ell^{(1)} / B_\ell^{(1)}, (1,k)]$ are all
of type $\cW(2,4,6,\ldots,2\ell,\ell+1/2)$, which is the spin content
that one would get by applying DS reduction to the superalgebra
$B(0,\ell)^{(1)}$ rather than to $B_\ell^{(1)}$ itself.

\vfil\eject
%\end


% chfront.tex
%---------------frontpage-----------------------------------------
\baselineskip=.8\baselineskip
\line{\hfill CERN-TH.6583/92}
\line{\hfill ITP-SB-92-23}
\vskip1cm

\centerline{\bf $\cW$-SYMMETRY IN CONFORMAL FIELD THEORY}
\vskip1.5cm

\centerline{{Peter Bouwknegt}\footnote{$^*$}{email:
bouwkneg@cernvm.cern.ch}}\smallskip

\centerline{\sl CERN - Theory Division}
\centerline{\sl CH-1211 Gen\`eve 23}
\centerline{\sl SWITZERLAND}
\vskip1cm

\centerline{{Kareljan Schoutens}\footnote{$^{**}$}{email:
schouten@max.physics.sunysb.edu}}\smallskip

\centerline{\sl Institute for Theoretical Physics}
\centerline{\sl State University of New York}
\centerline{\sl Stony Brook, NY 11794-3840}
\centerline{\sl USA}
\vskip2cm

\centerline{\bf Abstract}\smallskip

We review various aspects of $\cW$-algebra symmetry in two-dimensional
conformal field theory and string theory.
We pay particular attention to the construction of $\cW$-algebras
through the quantum Drinfeld-Sokolov reduction and through the coset
construction.
\vskip1cm

\centerline{To be published in: {\sl Physics Reports}}

\vfill
\line{CERN-TH.6583/92\hfil}
\line{ITP-SB-92-23\hfil}
\line{July 1992\hfil}
\baselineskip=1.25\baselineskip
\eject

% chrefs.tex, last updated 12/22/92 (3 items added)
%\pageno=123
%\writelabels
%
% This file contains a list of references in alphabetical order.
%
% The principal rule for label-assignment is that the label consists of
% the combination of (upper-case) first letters of each authors' surname,
% to which (lower-case) letters (a,b,c,...) are appended if there are
% several papers with the same author(s). In the case of one author or
% other confusing cases an additional (lower-case) letter is added.
%
%
\def\NP{Nucl. Phys.\ }
\def\PL{Phys. Lett.\ }
\def\CMP{Comm. Math. Phys.\ }
\def\PR{Phys. Rev.\ }
\def\IJMP{Int. Journ. Mod. Phys.\ }
\def\MPL{Mod. Phys. Lett.\ }
\def\AP{Ann. Phys.\ }
\def\PRL{Phys. Rev. Lett.\ }
\def\IM{Inv. Math.\ }
\def\AnMa{Ann. Math.\ }
\def\LMP{Lett. Math. Phys.\ }

\def\em{\it}
\def\sW{{\cal SW}}


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%  {\it Extended Sugawara construction for the superalgebras
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%  {\it Extended Sugawara construction for the superalgebras
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%  \PR {\bf D40} (1989) 415.}
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%  and selection rules of operator algebras in conformal field
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%\bref\CG{
%  E. Cremmer and J.-L. Gervais, {\it The quantum group structure
%  associated with nonlinearly extended Virasoro algebras},
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%\bref\CPS{
%  C. Crnkovic, R. Paunov and M. Stanishkov, {\it Fusions of conformal
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%  P. Degiovanni, {\it $\ZZ/N\ZZ$ Conformal field theories},
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%  V.S. Dotsenko  and V.A. Fateev,
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%  G. Felder, K. Gawedzki and A. Kupiainen, {\it The spectrum of
%  Wess-Zumino-Witten models}, \NP {\bf B299} (1988) 355.}
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%  J.M. Figueroa-O'Farrill and E. Ramos, {\it Existence and uniqueness
%  of the universal $w$-algebra}, preprint KUL-TF-91/27.}
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%  J.M. Figueroa-O'Farrill and E. Ramos, {\it The classical limit
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%  D. Friedan,
%  {\it A new formulation of string theory},
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%
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%
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  Y. Saitoh, N. Sakai and N. Yugami, {\it Bosonic construction of
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%\bref\Sa{
%  R. Sasaki, {\it Notes on extended conformal algebras},
%  preprint RRK 88-21.}
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\bref\SYb{
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  \PL {\bf 227B} (1989) 387.}
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  K. Schoutens, {\it $O(N)$-extended superconformal field theory
  in superspace}, \NP {\bf B295} [FS21] (1988) 634.}
%\bref\Scb{
%  K. Schoutens, {\it Representation theory for a class of $so(N)$
%  extended superconformal operator algebras},
%  \NP {\bf B314} (1989) 519.}
\bref\Scc{
  K. Schoutens, {\it Extensions of conformal symmetry in two dimensional
  quantum field theory}, Univ. of Utrecht PhD thesis, 1989.}
\bref\SS{
  K. Schoutens and A. Sevrin, {\it Minimal super $\cW_N$ algebras
  in coset conformal field theories}, \PL {\bf 258B} (1991) 134.}
\bref\SSvNa{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, {\it Quantum BRST
  charge for quadratically nonlinear Lie algebras},
  \CMP {\bf 124} (1989) 87.}
\bref\SSvNb{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen,
  {\it A new gauge theory for $\cW$-type algebras},
  \PL {\bf 243B} (1990) 245.}
\bref\SSvNc{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen,
  {\it Covariant formulation of classical $\cW$-gravity},
  \NP {\bf B349} (1991) 791.}
\bref\SSvNca{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, {\it Covariant
  $w_\infty$ gravity and its reduction to $\cW_N$ gravity},
  \PL {\bf 251B} (1990) 355.}
\bref\SSvNd{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, {\it Quantum
  $\cW_3$ gravity in the chiral gauge}, \NP {\bf B364} (1991) 584.}
\bref\SSvNe{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, {\it On the effective
  action of chiral $\cW_3$ gravity}, \NP {\bf B371} (1992) 315.}
\bref\SSvNf{
  K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, {\it Induced gauge
  theories and $\cW$-gravity}, in Proc. of `Strings and Symmetries 1991',
  Stony Brook, May 1991, ed.\ N. Berkovits {\it et al.}\
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\bref\Sc{
  S. Schrans, {\it Extensions of conformal invariance in two-dimensional
  quantum field theory}, Univ. of Leuven PhD thesis, 1991.}
%\bref\ScS{
%  A. Schwimmer and N. Seiberg, {\it Comments on the $N=2,3,4$
% superconformal algebras in two dimensions}, \PL {\bf 184B} (1987) 191.}
\bref\Seg{
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  groups}, Comm. Math. Phys. {\bf 80} (1981) 301.}
%\bref\Se{
%  A. Sevrin, {\it Superconformal algebras and supersymmetric
%  nonlinear $\sigma$-models}, Univ. of Leuven PhD thesis, 1988.}
\bref\STPS{
  A. Sevrin, W. Troost and A. Van Proeyen, {\it Superconformal
  algebras in two dimensions with $N=4$}, \PL {\bf 208B} (1988) 447.}
\bref\Sh{
  X. Shen, {\it $\cW$-infinity and string theory},
  preprint CERN-TH.6404/92.}
\bref\SY{
  J. Soda and H. Yoshii, {\it Kac formulas for the extended Virasoro
  algebras}, preprint RRK-88-5.}
\bref\SoS{
  G.M. Sotkov and M. Stanishkov, {\it Affine Geometry and
  $\cW_n$-gravities}, \NP {\bf B356} (1991) 439.}
\bref\SSZ{
  G.M. Sotkov, M. Stanishkov and C. Zhu, {\it Extrinsic Geometry
  of Strings and $\cW$-gravities}, \NP {\bf B356} (1991) 245.}
\bref\Su{
  H. Sugawara, {\it A field theory of currents}, \PR {\bf 170}
  (1968) 1659.}
\bref\Thi{
  K. Thielemans, {\it A Mathematica package for computing operator
  product expansions}, \IJMP {\bf C2} (1991) 787.}
\bref\Tha{
  J. Thierry-Mieg, {\it BRS-analysis of Zamolodchikov's spin 2
  and 3 current algebra}, \PL {\bf 197B} (1987) 368.}
\bref\Thb{
  J. Thierry-Mieg, {\it Generalizations of the Sugawara Construction},
  in `Nonperturbative
  Quantum Field Theory', ed. G. 't Hooft {\it et al.},
  Proc. Cargese School 1987 (Plenum Press, New York, 1988), 567.}
\bref\Tho{
  C. Thorn, {\it Computing the Kac determinant using dual model
  techniques and more about the no-ghost theorem}, \NP {\bf B248}
  (1984) 551.}
\bref\TV{
  T. Tjin and P. Van Driel, {\it Coupled WZNW-Toda models and covariant
  KdV hierarchies}, preprint ITFA-91-04.}
%\bref\Ty{
%  I.V. Tyutin, unpublished (1975).}
%\bref\Va{
%  C. Vafa, {\it Toward classification of conformal theories},
%  \PL {\bf 206B} (1988) 421.}
%\bref\Var{
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%  representations} (Prentice-Hall inc., 1974).}
\bref\Varn{
  R. Varnhagen, {\it Characters and representations
  of new fermionic $\cW$-algebras}, BONN-HE-91-16.}
\bref\Vea{
  E. Verlinde, {\it Fusion rules and modular transformations in
  2D conformal field theory}, \NP {\bf B300} [FS22] (1988) 360.}
%\bref\Veb{
%  E. Verlinde, {\it Conformal field theory and its application to string
%  theory}, Univ. of Utrecht PhD thesis, 1988.}
\bref\Wak{
  M. Wakimoto, {\it Fock representations of the affine Lie algebra
  $A_1^{(1)}$}, \CMP {\bf 104} (1986) 605.}
\bref\Waa{
  G.M.T. Watts, {\it Determinant formulae for extended algebras in two
  dimensional conformal field theory}, \NP {\bf B326} (1989) 648;
  erratum \NP {\bf B336} (1990) 720.}
\bref\Wab{
  G.M.T. Watts, {\it $\cW B$ algebra representation theory},
  \NP  {\bf B339} (1990) 177.}
\bref\Wac{
  G.M.T. Watts, {\it $\cW$-algebras and coset
  models}, \PL {\bf 245B} (1990) 65.}
\bref\Wad{
  G.M.T. Watts, {\it Extended algebras in conformal field theory},
  Trinity college PhD thesis, 1990.}
\bref\Wae{
  G.M.T. Watts, {\it $\cW B_n$ symmetry, Hamiltonian reduction and
  $B(0,n)$ Toda theory}, \NP {\bf B361} (1991) 311.}
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  Ward identities}, \PL {\bf 37B} (1971) 95.}
\bref\Wia{
  E. Witten, {\it Nonabelian bosonization}, \CMP {\bf 92} (1984) 455.}
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  of two-dimensional gravity}, \NP {\bf B340} (1990) 281.}
\bref\Wiab{
  E. Witten, {\it Topological sigma models},
  \CMP {\bf 118} (1988) 411.}
%\bref\Wib{
%  E. Witten, {\it Quantum field theory and the Jones polyniomial},
%  \CMP {\bf 121} (1989) 351.}
\bref\Wic{
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  Phys. Rev. {\bf D44} (1991) 314.}
\bref\Wid{
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\bref\WYa{
  Y.-S. Wu and F. Yu, {\it Hamiltonian structure, (anti-)self-adjoint
  flows in KP hierarchy and the $\cW_{1+\infty}$
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  Y.-S. Wu and F. Yu, {\it Nonlinearly deformed
  $\widehat{W}_\infty$ algebra and second hamiltonian structure
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\bref\Yab{
  K. Yamagishi, {\it $\widehat{W}_\infty$ algebra is anomaly free
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  (1989) 323.}

% table of contents for PhysRep,.....chtoc.tex
% last updated 12/22/92

\def\ni{\noindent}
\def\regel{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\tab{\hskip 5mm}
\def\tabtab{\hskip 12mm}

\centerline{\bf $\cW$-Symmetry in Conformal Field Theory}

\vskip 3mm

\centerline{P. Bouwknegt and K. Schoutens}

\vskip 3mm

\tenpoint

\line{1. Introduction \regel 3}

\line{\tab 1.1. Extensions of conformal symmetry \regel 3}

\line{\tab 1.2. Studying extended symmetries \regel 6}

\line{\tab 1.3. Outline of the paper \regel 10}

\line{2. Preliminaries \regel 11}

\line{\tab 2.1. Conformal invariance: basic notions \regel 11}

\line{\tab 2.2. OPE's, normal ordered
products and associativity \regel 17}

\line{\tab 2.3. Auxiliary field theories \regel 20}

\line{\tabtab 2.3.1. Free fields \regel 20}

\line{\tabtab 2.3.2. Affine Kac-Moody algebras and WZW models \regel 23}

\line{3. $\cW$-algebras and Casimir algebras \regel 28}

\line{\tab 3.1 $\cW$-algebras: definitions and
               the example of $\cW_3$ \regel 28}

\line{\tab 3.2 Casimir algebras \regel 33}

\line{\tab 3.3 $\cW$-superalgebras;
the example of super-$\cW_3$ \regel 35}

\line{4. $\cW$-algebras and RCFT \regel 38}

\line{\tab 4.1 The chiral algebra in RCFT's \regel 38}

\line{\tab 4.2 Examples: $\cW_3$ and super-$\cW_3$ minimal
models \regel 40}

\line{5. Classification through direct construction \regel 44}

\line{\tab 5.1. The method \regel 44}

\line{\tab 5.2. Overview of results \regel 45}

\line{\tabtab 5.2.1. Generic, linear algebras \regel 46}

\line{\tabtab 5.2.2. Generic, non-linear algebras \regel 48}

\line{\tabtab 5.2.3. Exotic algebras \regel 51}

\line{\tab 5.3. Relating various $\cW$-algebras \regel 52}

\line{\tabtab 5.3.1. Factoring out spin-${1 \over 2}$ fermions \regel 52}

\line{\tabtab 5.3.2. Twisted and projected $\cW$-algebras \regel 53}

\line{\tabtab 5.3.3. Relations with parafermion algebras \regel 54}

\line{\tabtab 5.3.4. $\whW_\infty$ as a universal structure \regel 55}

\line{6. Quantum Drinfeld-Sokolov reduction \regel 57}

\line{\tab 6.1. Introduction \regel 57}

\line{\tab 6.2. Lagrange approach:
                constrained WZW and Toda field theories \regel 58}

\line{\tab 6.3. Algebraic approach to DS reduction \regel 60}

\line{\tabtab 6.3.1. $\cW$-algebras from DS reduction \regel 60}

\line{\tabtab 6.3.2. Character technique \regel 68}

\line{\tabtab 6.3.3. Examples \regel 71}

\line{\tab 6.4. Representation theory \regel 79}

\line{\tabtab 6.4.1. The Kac determinant \regel 79}

\line{\tabtab 6.4.2. Completely degenerate representations
            and minimal models \regel 81}

\line{\tabtab 6.4.3. Character formulae \regel 83}

\line{7. Coset constructions \regel 85}

\line{\tab 7.1. Introduction \regel 85}

\line{\tab 7.2. Casimir algebras \regel 87}

\line{\tabtab 7.2.1. Generalities \regel 87}

\line{\tabtab 7.2.2. Character technique \regel 89}

\line{\tabtab 7.2.3. 3rd Order Casimir example \regel 91}

\line{\tab 7.3. $G \times G / G$ coset conformal field theories \regel 95}

\line{\tabtab 7.3.1. Deforming the singlet algebra \regel 95}

\line{\tabtab 7.3.2. Concrete example \regel 97}

\line{\tabtab 7.3.3. Representation theory \regel 99}

\line{\tabtab 7.3.4. The Kac determinant \regel 100}

\line{\tabtab 7.3.5. The limiting $\cW$-algebra of
                     diagonal coset models  \regel 101}

\line{\tab 7.4. Other cosets \regel 103}

\line{\tabtab 7.4.1. $G\times G'/G'$ coset conformal field theories
\regel 103}

\line{\tabtab 7.4.2. Dual coset pairs \regel 104}

\line{8. Further developments \regel 107}

\line{\tab 8.1. $\cW$-gravity \regel 107}

\line{\tab 8.2. $\cW$-symmetry in string theory \regel 111}


\line{9. Appendices \regel 116}

\line{\tab A. Lie algebra conventions \regel 116}

\line{\tab B. $\cW$-algebra nomenclature \regel 118}

\line{References \regel 120}

\vfil\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%               pkjmac.tex         last updated 06/16/91
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \input harvmac.tex    % version 9/91 required
%
%%
%%  Some additions/modifications to harvmac
%%
\global\newcount\mthno \global\mthno=1
\global\newcount\mexno \global\mexno=1
\global\newcount\mquno \global\mquno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
\global\subsecno=0\xdef\secsym{\the\secno.}\global\meqno=1\global\mthno=1
\global\mexno=1\global\mquno=1
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi  %if desired
\bigbreak\medskip\noindent{\bf\the\secno. #1}\writetoca{{\secsym} {#1}}
\par\nobreak\medskip\nobreak}
\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1\message{(\secsym\the
\subsecno. #1)}
\bigbreak\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
%
\def\appendix#1#2{\global\meqno=1\global\mthno=1\global\mexno=1%
\global\mquno=1
\global\subsecno=0
\xdef\secsym{\hbox{#1.}}
\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
%
%          theorems and examples and questions
%
\def\thm#1{\xdef #1{\secsym\the\mthno}\writedef{#1\leftbracket#1}%
\global\advance\mthno by1\wrlabeL#1}
\def\que#1{\xdef #1{\secsym\the\mquno}\writedef{#1\leftbracket#1}%
\global\advance\mquno by1\wrlabeL#1}
\def\exm#1{\xdef #1{\secsym\the\mexno}\writedef{#1\leftbracket#1}%
\global\advance\mexno by1\wrlabeL#1}
%
%     \ref\label{text}
% generates a number, assigns it to \label, generates an entry.
% To list the refs on a separate page,  \listrefs
%            on the same page,  \listrefsnoskip
%
\def\ref{\the\refno\nref}
\def\nref#1{\xdef#1{\the\refno}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=\jobname.refs\fi
\global\advance\refno by1\chardef\wfile=\rfile\immediate
\write\rfile{\noexpand\item{[#1]\ }\reflabeL{#1\hskip.31in}\pctsign}
\findarg}
\def\bref{\nref}
%
\def\listrefsnoskip{\footatend\immediate\closeout\rfile\writestoppt
\baselineskip=14pt{\bigskip\noindent {\bf  References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}
\nonfrenchspacing}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%    Here our macros start
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\p{\partial}
%
%  Greek abbreviations (to be used within $....$)
%
\def\al{\alpha} \def\be{\beta} \def\ga{\gamma}  \def\Ga{\Gamma}
\def\de{\delta}  \def\De{\Delta} \def\ep{\epsilon}
\def\vep{\varepsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta}
\def\Th{\Theta}  \def\vth{\vartheta} \def\io{\iota} \def\ka{\kappa}
\def\la{\lambda} \def\La{\Lambda} \def\rh{\rho} \def\si{\sigma}
\def\Si{\Sigma} \def\ta{\tau} \def\up{\upsilon}  \def\Up{\Upsilon}
\def\ph{\phi}  \def\Ph{\Phi}  \def\vph{\varphi} \def\ch{\chi}
\def\ps{\psi}  \def\Ps{\Psi} \def\om{\omega}  \def\Om{\Omega}
%
%  boldface abbreviations  (to be used within $..$)
%
\def\bC{{\bf C}} \def\bN{{\bf N}} \def\bQ{{\bf Q}} \def\bR{{\bf R}}
\def\bZ{{\bf Z}}
%
\def\bGa{{\bf\Ga}} \def\bLa{{\bf\La}}
%
% Calligraphic abbreviations (to be used within $..$)
%
\def\cA{{\cal A}} \def\cB{{\cal B}} \def\cF{{\cal F}} \def\cI{{\cal I}}
\def\cL{{\cal L}} \def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}}
\def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cZ{{\cal Z}}
\def\cC{{\cal C}} \def\cH{{\cal H}}
%
%
%  definition of \underhook{...}
%
\def\lefthook{{\vrule height5pt width0.4pt depth0pt}}
\def\righthook{{\vrule height5pt width0.4pt depth0pt}}
\def\leftrighthookfill{$\mathsurround=0pt \mathord\lefthook
     \hrulefill\mathord\righthook$}
\def\underhook#1{\vtop{\ialign{##\crcr$\hfil\displaystyle{#1}\hfil$\crcr
      \noalign{\kern-1pt\nointerlineskip\vskip2pt}
      \leftrighthookfill\crcr}}}
%
%  other abbreviations
%
\def\dCo{{\rm h}^{\vee}}
\def\Box{\hbox{\rlap{$\sqcup$}$\sqcap$}}

\def\hb{\hfill\break}
\def\ie{{\it i.e.\ }}
\def\eg{{\it e.g.\ }}
\def\eq#1{\eqno{(#1)}}

\def\proof{\noindent {\it Proof:}\ }
\def\example{\noindent {\it Example:}\ }

\def\ZZ{Z\!\!\!Z}           %%%%%%%%%%%%%
\def\NN{I\!\!N}             %
\def\RR{I\!\!R}             % produces reasonably looking math. Z,N,etc.
\def\CC{I\!\!\!\!C}         %
\def\QQ{I\!\!\!\!Q}         %%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% some Lie algebra macros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\bfg{{\bf g}}
\def\bfh{{\bf h}}
\def\bfnm{{\bf n}_-}
\def\bfnp{{\bf n}_+}
\def\hg{\hat{\bf g}}
\def\bft{{\bf t}}
\def\bfk{{\bf k}}
\def\whg{\widehat{\bf g}}
\def\whk{{\widehat{\bf k}}}
\def\rhv{{\rh^\vee}}
\def\Lap{\La^{(+)}}
\def\Lam{\La^{(-)}}

%%%%%%%%%%%%%%%%%%%%%%
\def\hz{\hat{z}}
\def\dhz#1{{\p\over \p \hat{z}_{#1} }}
\def\dz#1{{\p\over \p z_{#1} }}
\def\bs{\bigskip}
\def\ss{\smallskip}
\def\hr{\bs\centerline{\vbox{\hrule width5cm}}\bs}
\def\Homg{{\rm Hom\,}_{\cU(\bfg)}}    % Use within $..$
\def\Homnm{{\rm Hom\,}_{\cU(\bfnm)}}
\def\Homnp{{\rm Hom\,}_{\cU(\bfnp)}}
\def\vacL{|\La\rangle}
\def\vacLp{|\La'\rangle}
\def\nest#1{[\![\, #1\, ]\!]}
\def\aliv{\al_i^\vee}
\def\alp{\al_+}


%%%
%%%
%%% some macros for making commutative diagrams
%%%
\def\v{\vskip .5cm}
\def\Z{{\bf Z}}
\def\comdiag{
\def\normalbaselines{\baselineskip15pt
\lineskip3pt \lineskiplimit3pt }}
\def\mape#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\maprightunder#1{\smash{\mathop{\longrightarrow}\limits_{#1}}}
\def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{\,\,\,\, #1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{
                \hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapuright{\nearrow}
\def\mapdright{\searrow}
\def\mapuprightdown#1{\smash{\mathop{\nearrow}\limits_{\,\,\,\,\,\,#1}}}
\def\mapne#1{\smash{\mathop{\nearrow}\limits_{\,\,\,\,\,\,#1}}}
\def\mapupright#1{\smash{\mathop{\nearrow}\limits^{#1\,\,\,\,\,\,}}}
\def\mapupleft#1{\smash{\mathop{\nwarrow}\limits^{#1\,\,\,\,\,\,}}}
\def\mapdownright#1{\smash{\mathop{\searrow}\limits_{#1\,\,\,\,\,\,}}}
\def\mapse#1{\smash{\mathop{\searrow}\limits_{#1\,\,\,\,\,\,}}}
\def\mapupleftu#1{\smash{\mathop{\nwarrow}\limits_{\,\,\,\,\,\,#1}}}
\def\mapdownrightup#1{\smash{\mathop{\searrow}\limits^{\,\,\,\,\,\,#1}}}
\def\mapdownleft#1{\smash{\mathop{\swarrow}\limits^{#1\,\,\,\,\,\,}}}
\def\smapdown#1{\big\downarrow\rlap{$\vcenter{
                \hbox{$\scriptstyle#1$}}$}}
\def\mapcr#1#2{\smash{\mathop{\nearrow\llap{$
\hbox{$\searrow\,\,$}$}}
                  \limits^{\,\,\,\,\,\,#1}_{\,\,\,\,\,\,#2}}}
\def\mapcrossnar#1#2{\smash{\mathop{\nearrow\llap{$
\hbox{$\searrow\,\,$}$}}
                  \limits^{\,\,\,\,#1}_{\,\,\,\,#2}}}

\def\kpfh#1{S^{w(#1)}_{\La,\La'}}
\def\kpfg#1{F^{w(#1)}_{\La}}
\def\kpcoh#1#2#3{H^{#1}_{#2}(#3)}
\def\hr{\medskip\centerline{\vbox{\hrule width5cm}}\medskip}
\def\kphom#1#2{{\rm Hom}_{{\cU}(#1)}(#2)}
\def\state#1{|#1\rangle}
\def\kpmgh#1{#1\otimes F_{gh}}
\def\kpdeg#1{{\rm deg}(#1)}
\def\kpfock#1#2{F^{#1}_{#2}}
\def\con#1{[\![\,#1\,]\!]}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%    some additional macros for ch67
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\fhg{{{\hbox{{\hbox{$\scriptstyle\bf g$}}\kern-.8em\lower.5ex%
\hbox{$\scriptstyle\widehat{\phantom{\bf g}}$}}}}}
\def\NO#1{:\!{#1}\!:}
\def\cite#1{[#1]}
\def\onethird{\textstyle{{1\over3}}}
\def\zb{\bar{z}}
\def\del{\partial}
\def\delb{\bar\partial}

\def\wLa{{\widehat{\La}}}

\def\third{{\textstyle{1\over3}}}
\def\threehalf{{\textstyle{3\over2}}}

\def\whT{\widehat{T}}

%%%%%%%%%%%some additional used in ch1234589

\def\sc{\tenrm}

\def\sW{{\cal SW}}
\def\frac#1#2{{#1\over #2}}  % LaTeX remnant

\def\zw{(z-w)}
\def\Th{\widehat{T}}
\def\Wh{\widehat{W}}
\def\del{\partial}
\def\hbar{\bar{h}}
\def\htil{\dCo}
\def\zb{\bar{z}}
\def\pr{\prime}
\def\prpr{\prime\prime}
\def\prprpr{\prime\prime\prime}
\def\kb{\bar{k}}
\def\Lb{\overline{L}}
\def\whh{\widehat{\bf h}}
\def\whso{\widehat{so}}
\def\whsu{\widehat{su}}
\def\whW{\widehat{\cW}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
%%%%   END MACRO FILE
%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


