\newsec{Equivariant Cohomology}
\seclab\sEQCO

In this chapter we begin developing some of the crucial mathematical background
for topological field theory. 
We begin with some of the algebraic structures in topological field theory.
These, in turn, are related to deep constructions in topology. 
The reason for reviewing this material  is that in all examples
{\it the BRST operator of a topological field theory is the differential for a model of
$\CG$-equivariant cohomology of a space of fields.}
\foot{In some models $\CG$ is the trivial group.}

\subsec{Classifying Spaces}

Let $G$ be a group and $\lieg$ its Lie algebra. 
Some of what follows is only rigorously true when
$G$ is compact, but the formal discussion
 can be applied to any group. In particular, 
in topological field theory, it is applied to infinite dimensional groups. 


A $G$-manifold $M$ has an action $x\to g \cdot  x$, for all
$x\in M$ and $g\in G$.
The action of $G$ is said to be {\it free} if, for any $x\in M$,
%
\eqn\free{
g \cdot x ~=~ x  ~\Longleftrightarrow~ g ~=~ 1}
%
that is, there are no nontrivial isotropy groups.
If the action of $G$ is free on $M$, then the quotient space $M / G$
forms the base space of a principal $G$ bundle
$$\matrix{
           M             & \mapleft{} & G \cr
\mapdown{\pi} &                      &    \cr
       M / G           &                     &    \cr}
$$
where the quotient space is smooth.
In many cases of interest to physics and mathematics, the group
action is not free.
This leads to the considerations of \sCtoOM\ below. 

\vskip0.1truein\noindent
{\bf Definition}:
To a group $G$ we can associate the {\it universal $G$-bundle}, $EG$, which
is a very special space, satisfying: 
\item{1.}
$G$ acts on $EG$ without fixed points. 
\item{2.}
$EG$ is contractible.

\vskip0.1truein\noindent
{\bf Examples:}
\vskip0.15truein
\begintable
 $G$ | $EG$ | $BG$ \elt
 $\IZ$ | $\IR$ | $S^1$ \elt
 $\IZ^n$ | $\IR^n $ | $S^1 \times \cdots\times S^1$ \elt
 $U(1)=SO(2)$ | $S(H)=\lim_{n\to \infty} S^{2n+1}$ | $\IC P^\infty=\lim_{n\to \infty} \IC P^n$ \elt
 $U(k)$ | $V_k(H)$ | $G_k(H)$  \elt
$\CG$:local gauge transform | $\CA$: Yang-Mills potentials | $\CA/\CG$ \elt
 $\Diff(\Sigma)\times \Weyl(\Sigma)$ | $\MET(\Sigma)$ | $\CM_{h,0}$ \elt
Mod(h,0)  |  Teichm\"uller |  $\CM_{h,0}$
\endtable
\vskip0.15truein

The third and fourth rows are somewhat unfamiliar in physics, 
but the last 3 rows are quite familiar. 
%($Gr_0(H)$ is one version of 
%the universal  Grassmannian\refs{\PrSe} that plays a role in many 
%investigations related to free fermions.)  
%
Evidently, classifying spaces play an important role in physics.  

\vskip0.1truein\noindent
{\bf Remarks:}
\item{1.}
$EG$ is the ``platonic $G$-bundle."  {\it Any} $G$ bundle is a pullback: 
%
\eqn\pullback{\matrix{
P\cong f^\ast EG &  & EG\cr
     \mapdown{}     &  & \mapdown{}\cr
              M              & \mapright{f} & BG\cr}}
%
That is, we can find a copy of any conceivable $G$-bundle 
sitting inside $EG\to BG$ \husemoller.
Moreover, isomorphism classes of bundles 
are in 1-1 correspondence 
with homotopy classes, $[ f\colon M \to BG]$.
\item{2.}
$BG$ is unique up to homotopy type. $EG$ is unique up to 
``equivariant homotopy type."  Recall that two spaces 
$X$ and $Y$ have the same homotopy type if there are maps 
$f\colon X\to Y$ and $g\colon Y\to X$ with $fg$ and $gf$ both 
homotopic to $1$. The maps are said to be 
{\it equivariant} if they commute with the $G$-action.
\item{3.}
There is an explicit combinatorial construction of $EG$ for 
any topological group due to Milnor. See \refs{\husemoller,\steenrod}\
for a description.
\item{4.}
In the last three rows we have ignored an important subtlety, 
namely, that  there are still fixed points.  In the gravity case (last example)
these arise from Riemann surfaces with automorphisms. By 
restricting to diffeomorphisms which preserve $H^\bullet (\Sw,\IZ_3)$ 
one can eliminate all fixed points \bers.
In the Yang-Mills case (third from last example) one must divide the gauge
group by its center (global $\IZ_N$ transformations, for $SU(N)$) and 
cut out the reducible connections described in chapter \sTYMT\ below.
The space of irreducible connections $\CA^{irr}$ is still 
contractible \refs{\AtBoym,\DoKro}. 

  
\exercise{The contractible sphere}

Note that, among other things, the above examples assert that 
the unit sphere in Hilbert space: 
%
\eqn\unsphhl{
\{(x_1,x_2,\dots )\in \ell_2(\IR)~ \vert~  \sum x_i^2=1 \}}
%
is contractible! 
Prove this by first showing that the ``Hilbert hotel map"
%
\eqn\hilhotl{
(x_1,x_2,\dots )\to 
(0,x_1,x_2,\dots )}
%
is homotopic to 1. Then give a deformation retract of the 
``equator" $\{ \vec x\colon x_1=0\} $ to the ``north pole",
$(1,0,\dots)$.

\endexercise


\subsec{Characteristic classes}
\subseclab\sCC

Let us review briefly some of the theory of  characteristic classes. 
Although $EG$ is contractible, it is a nontrivial 
bundle over $BG$. 
{\it Characteristic classes} are elements of the 
cohomology $H^\bullet (BG)$ which measure the twisting of 
the bundle. By the universal property, characteristic 
classes pull back to all $G$-bundles and measure 
twisting. Indeed, in a sense, all natural ways of measuring
the topology of $P\to M$ are obtained by pullback from $H^\bullet (BG)$.
(Making this statement precise would take us into category theory.)

Characteristic classes of $P\to M$  are formed from the field strengths
$F\in \Omega^2 ( P, \lieg)$ of a connection $A\in \Omega^1 ( P, \lieg)$ on $P$. 
These satisfy:
%
\eqn\unvrls{\mathboxit{
\eqalign{dA &= F-\half [ A, A]\cr
dF &= -[ A, F]\cr}}}
%
Let us form 
$$
\widetilde{ch_n}={1\over n!(2 \pi i)^n } \Tr~ F^n\in \O^{2n}(P)
$$
Using \unvrls\ one easily shows that this is a closed form on $P$ (the same is true
for $\CP(F)$, if $\CP$ is any invariant form on the Lie algebra $\lieg$).
Since $EG$ is contractible, its cohomology is trivial.
Indeed $\widetilde{ch_n}$ is exact:
$$\eqalign{
\widetilde{ch_n}
=& d \omega_{0,n}\cr
=& d \left ( {1\over{(n - 1)! (2 \pi i )^n}} \sum_{i=0}^{n-1}
{1\over{(n+i)}} \Tr[( d A )^{n-i-1} ( A )^{2i + 1} ] \right ). \cr}
$$
To get interesting cohomology we must discuss closed forms on $BG$.
This leads to the notion of {\it basic forms}.

\ifig\puonei{Principal $U ( 1 )$ bundle over $M$.}
{\epsfxsize3.5in\epsfbox{PrinBun.eps}}

\subsubsec{Basic Forms}


Recall from \sssVTV\  that a principal bundle $P$ has an action of $X\in \lieg$ on $P$
via the vertical vector fields $\xi(X)$ (we will drop the $\xi$ in our notation.)
For example, for a principal $U(1)$ bundle the action looks like \puonei.
There are {\it two}  associated actions of $\lieg$ on the differential forms $\O (P)$:
%
\eqn\twoacts{\eqalign{
\iota (X)\colon \O^{k} ( P ) \to \O^{k-1}  ( P ) & \qquad {\rm contraction} \cr
\CL(X) \equiv [\iota (X),d]_+ :\O^k ( P ) \to \O^k ( P ) & \qquad {\rm Lie\ derivative} \cr}}
%

Forms $\widetilde \omega$ on $P$ which are of the form $\pi^*(\omega)$ 
for $\omega \in\O (M)$ are
called ``basic".  Such forms are characterized by 
%
\eqn\bscfrms{\eqalign{
\iota (X) \widetilde \omega & = 0 \qquad\quad {\rm no\ vertical\ components}\cr
\CL(X) \widetilde \omega & = 0 \qquad\quad {\rm no\ vertical\ variation}.  \cr}}
%

It is easily checked that $\widetilde{ch}_n$ are basic, so that 
$\widetilde{ch}_n = \pi^*(ch_n) $.
The forms $ch_n$ define nontrivial cohomology classes on $BG$. 
It can be shown that the cohomology class is independent of the choice of
connection $A$.  

\vskip0.1truein\noindent
{\bf Remark}:
We have defined these classes using differential forms.
In fact they can be defined purely topologically as elements of the 
{\it integral cohomology} $H^\bullet (BG;\IZ)$ \milnor.

\subsec{Weil Algebra}
\subseclab\ssWA

We now introduce an algebraic analog of $EG$. 
We may view a connection as a map from the 
dual of the Lie algebra, $\lieg\dual$, to a differential form:
%
\eqn\weilhomi{\eqalign{
A\in \Omega^1(P,\lieg) \qquad & \qquad  
\longleftrightarrow\qquad A\colon\lieg\dual \to \Omega^1(P)\cr
F\in \Omega^1(P,\lieg) \qquad & \qquad  
\longleftrightarrow \qquad  F\colon\lieg\dual \to \Omega^2(P)\cr}}
%
where $\lieg\dual $ is the dual to $\lieg$. 
Let us return to \unvrls.
The reinterpretation \weilhomi\ motivates the 
\vskip0.1truein\noindent
{\bf Definition}: The Weil algebra of $\lieg$ is 
the differential graded algebra (DGA) 
$$\CW ( \lieg ) = S ( \lieg\dual~ ) \otimes \Lambda ( \lieg\dual~ )$$
where $S( \cdot )$ is the symmetric algebra and
$\Lambda ( \cdot )$ is the exterior algebra.
It may be described in terms of generators by choosing 
a basis $\lieg=\{ T_i \}$ and taking generators
for  $S( \lieg \dual ~)$
given by $\{ \phi^i \}_{i=1,\ldots, \dim~ G}$ of degree 2
and generators for $\Lambda ( \lieg\dual~ )$ given by
$\{ \theta^i \}_{i=1,\dots, \dim~ G}$, of degree 1. 
The Weil algebra becomes a differential algebra upon
introducing  the differential:
%
\eqn\WeilDeriv{\eqalign{
d_{\CW}  \theta^i ~=&~ \phi^ i -\half f^i_{~ jk}  \theta^j  \theta^k
                  \cr
d_{\CW} \phi^i   ~=&~ - f^i_{~ jk}   \theta^j\phi^k\cr}}
%
where $f^i_{~ jk}$ are the structure constants of $\lieg$.
It may be seen that $d_{\CW}$ is nilpotent,
$d_{\CW}^2 = 0 $. 
 
\vskip0.1truein\noindent
{\bf Remarks:} 

\item{1.} One could {\it define}  
a connection on $P$ as a homomorphism $\CW(\lieg)\to \Omega(P)$. 

\item{2.} 
The Weil algebra may be thought of ``more 
physically" in terms of ``$b,c,\beta,\gamma$ systems"
as follows. 
The space $S(\lieg\dual~)$ may be identified with 
the space of functions on the Lie algebra. 
We denote generators of the polynomial 
functions  (with respect to the 
basis $T_i$) by $\gamma^i$, of degree two. 
Similarly $\CW(\lieg)$ may be identified with the space of functions on a superspace
\foot{The $\Pi$ indicates that the fiber is considered 
odd. See, for example, \manin.  }
 built
from the tangent bundle of $\lieg$,  $\hat \lieg=\Pi T\lieg$. 
The odd generators of functions on the tangent fibers are denoted by $c^i$.
So a function on the superspace $\hat \lieg$ is a superfield
$$
\Phi(\gamma^i, c^i). 
$$
This space of functions is graded.
The grading is referred to as ``ghost number" in physics. 
Now we introduce an algebra of operators on the space of functions: 
%
\eqn\comrels{
[\beta_i , \gamma^j] = \delta_i^j \qquad\qquad 
\{ b_i, c^j\} = \delta_i^j}
%
The action of the operator:
%
\eqn\bcbetgam{
d_{\cal W} \to Q= - f^i_{jk} c^j \gamma^k \beta_i  - 
\half f^i_{jk} c^j c^k b_i +\gamma^i b_i}
%
coincides with the Weil differential. Note that $Q$ is very reminiscent of  the form of 
``BRST operators" for supersymmetric gauge principles.
We will comment further on this connection in section \ssECvLAC\ below. 

\subsubsec{Basic subcomplex}

The Weil algebra has  properties analogous to those of $EG$. 
Just as $EG$ is contractible, the cohomology of the Weil algebra 
is trivial:  
$$H^\bullet ( {\cal W} (\lieg),d_\CW) =\delta_{ \bullet, 0} \IR .$$ 

\exercise{Trivial cohomology}

Show that the cohomology is trivial by choosing 
a different set of generators for $W(\lieg)$. (Hint: Use the first 
equation to make one generator exact). 

\endexercise

To get interesting cohomology we take 
our cue from the theory of characteristic classes 
reviewed above and  introduce two differential operators on $\CW (\lieg )$:
the interior derivative, $I_i$, and Lie derivative, $L_i$, defined
by their actions on the generators:
%
\eqn\DefDeriv{\eqalign{
I_i  \theta^j ~=&~ \delta_i^j\cr
I_i \phi^j ~=&~ 0\cr
L_i ~\colon=&~ [ I_i, d_\CW ]_+ \cr}}
%
$I_j$ has degree $-1$ while $L_j$ is of degree zero.
\vskip0.1truein\noindent
{\bf Definition}: An element  $\eta \in \CW ( \lieg )$ will be called
\item{a.} {\it Horizontal} if
$\eta \in \cap_{i=1}^{\dim G} \ker ( I_i  )$,
\item{b.} {\it Invariant} if
$\eta \in  \cap_{i=1}^{\dim G} \ker ( L_i )$,
\item{c.} {\it Basic} if $\eta$ is both horizontal and invariant.

\exercise{Action of $L_i$}
Show that $L_i$ are the generators of the 
co-adjoint action of $\lieg$ on $\lieg\dual$. 
\endexercise

We denote the basic subcomplex of $\CW(\lieg)$ by $B\lieg$. 
It is straightforward to calculate $H^\bullet ( B\lieg )$, since 
on the basic subcomplex $d_\CW$ is zero! Hence we 
need only determine explicitly $B\lieg$. Horizontality implies that 
we are in $S( \lieg\dual~ )$, and invariance translates into 
invariance under the coadjoint action of $\lieg$ on $\lieg\dual$. 
We denote the invariant elements under this $G$-action 
by $S(\lieg\dual~)^G$. 

To summarize, we have: 
$$\eqalign{
B\lieg  &= \CW(\lieg)_{\rm basic} \cr
H^\bullet (B\lieg)&= S(\lieg\dual~)^G \cr}
 $$
These are the invariant polynomials on the Lie algebra $\lieg$, 
i.e., $B\lieg$ is the algebra of Casimir invariants. 
The relation between $B\lieg$ and $BG$ is more than just an 
analogy:
\vskip0.1truein\noindent
{\bf Theorem \sEQCO.1}: If $G$ is a compact connected Lie group then 
$H^\bullet ( BG ) = H^\bullet ( B\lieg )=S( \lieg\dual~ )^G$.
\vskip0.1truein\noindent
For a proof see, for example, \AtBomm.
\vskip0.1truein\noindent
{\bf Example:} $G=SO(2)=U(1)$.
As we have already seen, the classifying space of $U ( 1 )$ is $\IC P^\infty$,
whose cohomology is a polynomial algebra on a single generator, $\Omega$,
of degree two.
Further $\lieg = u ( 1 )$ has a single generator, so that
$$
H^\bullet ( \IC P^\infty ) = S (u(1)\dual~ ) = \IC[ \O ]
$$ 
\vskip0.1truein\noindent
{\bf Remark}: 
Already in finite dimensions the theorem is not  true for noncompact groups.
A proper statement involves ``continuous cohomology"\refs{\Bo}. 
In topological field theory the relevant groups are infinite dimensional and certainly
not compact. 
Nevertheless, the cohomologies are remarkably close to those of related compact groups,
although, as we will see, they involve spacetime in an interesting way
(through the ``descent equations"). 

\subsec{Equivariant Cohomology of Manifolds}
\subseclab\sCtoOM

Suppose a manifold $M$ has a $G$-action.  
In general, there are fixed points and $M/G$ is not a manifold.  
It is difficult to discuss the cohomology in such situations.  
A standard trick in algebraic topology is to replace 
$$
M\to  EG\times M
$$
This has a free $G$-action so 
$$
EG\times_G M \equiv {EG\times M\over G},
$$
where $g\cdot (e,x)=(e g^{-1}, gx)$,  
is a manifold. On the other hand, $EG$ is contractible so $EG\times M \cong M$ in
homotopy theory.  
Note, if the $G$ action on $M$ is free then indeed $EG\times_G M\cong M/G$
in homotopy theory.
In general, we can regard $EG\times_G M$ as a bundle over $BG$ with fiber $M$. 

The above observations motivate the topological definition of equivariant cohomology:
\vskip0.1truein\noindent
{\bf Definition.}  The  topological $G$-equivariant cohomology of 
$M$ is
% 
\eqn\tpleqv{
H^\bullet_{G,{\rm topological}} (M) \equiv H^\bullet (EG\times_G M)}
%
Note that $H^\bullet _G(pt) = H^\bullet (BG)$ is highly nontrivial!

As in our previous discussion, there is a corresponding algebraic description.
Algebraically, the replacement  $M\to EG\times M$ is analogous to 
$$
\O( M ) \to \CW( \lieg ) \otimes \O ( M )
$$
We must define {\it basic forms}.
Let $X_i$ be the vector fields on $M$ corresponding to the action of $T_i\in\lieg$. 

\vskip0.1truein\noindent
{\bf Definition}: Elements of $\eta \in \CW ( \lieg ) \otimes \Omega ( M )$
will be called
\item{a.} {\it Horizontal} if
$\eta \in \cap_{i=1}^{\dim G}\ker ( I_i \otimes 1 + 1 \otimes \iota ( X_i ) )$,
\item{b.} {\it Invariant} if
$\eta \in \cap_{i=1}^{\dim G}\ker ( L_i \otimes 1 + 1 \otimes \cL(X_i) )$,
\item{c.} {\it Basic} if $\eta$ is both horizontal and invariant.
\vskip0.1truein\noindent
%
%(where again we have suppressed the canonical isomorphism
%$\xi\colon \lieg \to ( T P )^{\rm vert}$ in our notation).
%
\vskip0.1truein\noindent
{\bf Definition}: 
The algebraic $G$-equivariant cohomology of $M$ is 
$$
H^\bullet_{G,{\rm algebraic}} ( M)
\equiv H^\bullet ( (\CW(\lieg) \otimes \O(M))_{\rm basic}, d_T)
$$
where
%
\eqn\totdiffl{
d_T = d_\CW\otimes 1 + 1\otimes d}
%
is the differential. 

Analogous to Theorem \sEQCO.1\ above, we have:
\vskip0.1truein\noindent
{\bf Theorem \sEQCO.2 } For $G$ compact 
$$
H^\bullet_{G,{\rm topological}} (M)= H^\bullet_{G,{\rm algebraic}} ( M)
$$
Again, for further details, see \AtBomm. 
\vskip0.1truein\noindent
{\bf Example}:
One of the most important cases for us is when $G=\Diff(\Sigma)\times \Weyl$, then,
according to our table,  $H^\bullet_G ( pt )=H^\bullet (\CM_{h,0})$.
Clearly, $G$-equivariant cohomology is related to 2D topological gravity.
If we let $M$ be the configuration space of a sigma model, $\MAP ( \Sw, X )$,  then
$G$-equivariant cohomology is related to 2d topological gravity coupled to a sigma model. 
The fact that $H^\bullet_{S^1} ( pt )=\IC[ \Omega ]$ is a polynomial algebra (with elements
of arbitrarily high degree = ghost number) is probably related to the ``special states" of
$D\leq 2$ string theory \lz. 

\subsec{Other formulations of equivariant cohomology}

This section follows \Kal.
It is typical in mathematics that a given object, cohomology 
groups for example, can be characterized or formulated in 
many very different ways. This is true of equivariant 
cohomology. 
There are three algebraic models commonly encountered 
in the literature.  The first model is the Weil model discussed above. 

\subsubsec{Cartan Model}
\subsubseclab\sssCartan

The extreme simplicity of $B\lieg$ suggests that a complex 
much simpler than the Weil complex suffices.
\vskip0.1truein\noindent 
$\bullet$ Complex: 
%
\eqn\cartcompl{
 S(\lieg\dual~) \otimes \O(M) \qquad .}
%
\vskip0.1truein\noindent
$\bullet$ Differential:
%
\eqn\CartanDiff{\eqalign{ 
d_{\sst \cC} \phi^i ~=&~ 0\cr
d_{\sst \cC} \eta   ~=&~ ( 1 \otimes d - \phi^i \otimes \iota_i ) \eta\cr
&= (d-\iota_\phi) \eta\cr}}
%
for $\eta\in \O(M)$. 
Note that $d_\cC^2 = - \phi^i \otimes \CL_i $ and
in general, $d_\cC$  does not square to zero. On the invariant subcomplex
defined by:
%
\eqn\crtcplx{
\O_G(M)\equiv \bigl ( S ( \lieg\dual~ ) \otimes \Omega ( M ) \bigr )^G}
%
it does square to zero:  $d_\CC^2=\phi^i L_i\otimes 1 \to 0$.
Elements of $\O_G(M)$ are called {\it equivariant differential forms}. 

We will see
%
\eqn\spclcse{
H^\bullet \Bigl ( \bigl ( \CW ( \lieg ) 
\otimes \Omega ( M ) \bigr )_{\rm basic},
d_\cW \Bigr ) ~\equil~
H^\bullet \Bigl ( \bigl ( S ( \lieg\dual~ ) \otimes \Omega ( M ) \bigr )^G,
d_{\sst \cC} \Bigr )}
%
as a special case of a more general result. 

\subsubsec{BRST Model}  
\subsubseclab\sssBRSTM

In the physical context of topological field theories it turns out that another model of
equivariant cohomology arises naturally.
This is called the ``BRST model,'' or, sometimes, the ``intermediate model.'' 
As a vector space, the complex of the BRST model is identical to that of the Weil model:
$$
\cW ( \lieg ) \otimes \Omega ( M )
$$
but now the differential $d_B$ is:
%
\eqn\BRSTOp{
d_B ~=~ d_{ \CW} \otimes 1 + 1 \otimes d
    + \theta^i \otimes \cL_i - \phi^i \otimes \iota_i}
%
with $d_B^2=0$. 
As in the Weil model, the cohomology of this complex is trivial.
One must restrict to a subcomplex in order to calculate the equivariant cohomology. 
The analog of the basic subcomplex of the Weil model is in fact the subcomplex 
$ \bigl ( S ( \lieg\dual~ ) \otimes \Omega ( M ) \bigr )^G$ of the Cartan model.
On this subcomplex we can clearly  identify $d_\cC=d_B$ (and so $d_\cC^2 = 0$.) 

\subsubsec{Equivalences} 

It was shown by Kalkman that the BRST and Weil models of equivariant 
cohomology are related by the algebra automorphism of conjugation by 
$\exp( \theta^i \iota_i )$:
%
\eqn\algaut{
e^{\theta^i \iota_i} d_T e^{- \theta^i \iota_i} = d_B}
%
where $d_T$ is given by \totdiffl. 

\exercise{}

Prove \algaut\ by computing separately $e^{\Ad ~\theta^i \iota_i} d_\CW $
and $e^{\Ad ~\theta^i \iota_i} d $. 
Note that it follows immediately that $d_B^2=0$, something which is not 
manifest from \BRSTOp. 

\endexercise

Similar computations to those above show that 
%
\eqn\mpbsc{\eqalign{
e^{\Ad~ \theta^i \iota_i} [ I_i \otimes 1 + 1 \otimes \iota_i ]
&=  I_i \otimes 1\cr
e^{\Ad ~ \theta^i \iota_i}[ L_i \otimes 1 + 1 \otimes \cL_i ]
&= [ L_i \otimes 1 + 1 \otimes \cL_i ]\cr}}
%
from which it follows that the basic subcomplex of the Weil model is mapped to the
Cartan subcomplex in the BRST model.
This proves \spclcse. 

\vskip0.1truein\noindent
{\bf Remark:}
A very similar conjugation appears in the papers \refs{\echikann,\getzler}
which relate the ``string picture'' and the ``matter picture'' of topological string 
theory (in the canonical formalism).

\subsubsec{Axiomatic Formulation}

The definition of equivariant cohomology can be axiomatized. 
We are here following \getzler.
It is not difficult to see that the 
equivariant cohomology groups 
satisfy the following three properties: 

\item{1.} {\it Normalisation}: If the $G$-action is free, then
$H_{\sst G}^\bullet ( M ) \equil H^\bullet ( M / G )$.
\item{2.} {\it Homotopy Invariance}: If $f\colon M_1 \to M_2$ is
an equivariant map inducing a homotopy equivalence, then
$f^\ast \colon H_{\sst G}^\bullet ( M_2 )
\to H_{\sst G}^\bullet ( M_1 )$ is an isomorphism.
\item{3.} {\it Mayer-Vietoris}: If $M = U \cup V$, where $U$ and $V$
are invariant open submanifolds of $M$, then there is the long
exact sequence
$$\matrix{
                &                                       &
                &                                       
                                                        &
                &                                       &
                \cr
                & \cdots                                &
\longrightarrow & H_{\sst G}^{\bullet-1} ( U )
           \oplus H_{\sst G}^{\bullet-1} ( V )          &
\longrightarrow & H_{\sst G}^{\bullet-1} ( U \cap V )   &
\longrightarrow \cr
\longrightarrow & H_{\sst G}^\bullet ( M )              &
\longrightarrow & H_{\sst G}^\bullet ( U )
           \oplus H_{\sst G}^\bullet ( V )              &
\longrightarrow & H_{\sst G}^\bullet ( U \cap V )       &
\longrightarrow \cr
\longrightarrow & \cdots                                &
                &                                       &
                \cr}
$$

These three properties are all clear from the topological definition.
In fact, they serve to characterize the cohomology groups uniquely and thus serve as an 
axiomatic definition of equivariant cohomology. Technically, 
equivariant cohomology is a contravariant functor from the category of $G$-manifolds to the category of 
graded vector spaces, $H_{\sst G}^\bullet ( M )$. 

\subsec{Example 1: $S^1$ -Equivariant Cohomology}

\subsubsec{Point}

We have already seen $H_{S^1}^\bullet (pt) = \IC[\O]$, where 
$\O$ is of degree two. 

\ifig\twosph{Figure of $S^2$ with $U ( 1 )$ action.}
{\epsfxsize2.0in\epsfbox{Sphere.eps}}

\subsubsec{$S^2$}

Consider the standard $U(1)$ action on the two-sphere, \twosph.
It is interesting to compare the different formulations here. 
{}From the axiomatic point of view we calculate the equivariant cohomology
$H_{S^1}^\bullet ( S^2 )$ as follows.
Introduce the standard open covering of $S^2$ by two disks:
$U_1 = S^2 - \{ \infty \}$ and $U_2 = S^2 - \{ 0 \}$.
Then the Mayer-Vietoris long exact sequence for the equivariant
cohomology reads:
%
\eqn\MVEquiv{\matrix{
   0 \to        &         H_{\sst S^1}^0 ( S^2 )        &
\longrightarrow & H_{\sst S^1}^0 ( U_1 ) \oplus
H_{\sst S^1}^0 ( U_2 )                                  &
\longrightarrow & H_{\sst S^1}^0 ( U_1 \cap U_2 )       &
\longrightarrow \cr
\longrightarrow & H_{\sst S^1}^1 ( S^2 )                &
\longrightarrow & H_{\sst S^1}^1 ( U_1 )
           \oplus H_{\sst S^1}^1 ( U_2 )                &
\longrightarrow & H_{\sst S^1}^1 ( U_1 \cap U_2 )       &
\longrightarrow \cr
\longrightarrow & H_{\sst S^1}^2 ( S^2 )                &
\longrightarrow & H_{\sst S^1}^2 ( U_1 )
           \oplus H_{\sst S^1}^2 ( U_2 )                &
\longrightarrow & H_{\sst S^1}^2 ( U_1 \cap U_2 )       &
\longrightarrow \cr
\longrightarrow & \cdots                                &
                &                                       &
                \cr}}%
$S^1$ does not act freely on $S^2$ since the north and south poles
are fixed points.
$S^1$ does, however, have a free action on $U_1 \cap U_2$, so that
by the normalisation axiom,
$H_{S^1}^i ( U_1 \cap U_2 ) = H^i ( U_1 \cap U_2 / S^1 )$,
which vanishes for $i \ge 1$.
As a result \MVEquiv\ splits into subsequences:
%
\eqn\MVSub{\matrix{
    0 \to       &         H_{\sst S^1}^0 ( S^2 )        &
\longrightarrow & H_{\sst S^1}^0 ( U_1 ) \oplus
H_{\sst S^1}^0 ( U_2 )                                  &
\longrightarrow & H_{\sst S^1}^0 ( U_1 \cap U_2 )       &
\longrightarrow \cr
\longrightarrow & H_{\sst S^1}^1 ( S^2 )                &
\longrightarrow & H_{\sst S^1}^1 ( U_1 ) \oplus
H_{\sst S^1}^1 ( U_2 )                                  &
\longrightarrow &                      0                &
\longrightarrow \cr
\longrightarrow & H_{\sst S^1}^2 ( S^2 )                &
\longrightarrow & H_{\sst S^1}^2 ( U_1 )
           \oplus H_{\sst S^1}^2 ( U_2 )                &
\longrightarrow &                     0                 &
\longrightarrow \cr
\longrightarrow & \cdots                                &
                &                                       &
                \cr}}
%
For $i \ge 2$, we simply have
%
\eqn\ECSoSt{
H_{\sst S^1}^i ( S^2 )
~\equil~ H_{\sst S^1}^i ( U_1 ) \oplus H_{\sst S^1}^i ( U_2 ). }
%
Using the axiom of homotopy invariance, we see that the maps
$\varphi_i\colon U_i \to pt$ are equivariant maps which induce
homotopy equivalences, so that $H^0=\IC$ and hence:
%
\eqn\ECSoSt{
H_{\sst S^1}^\bullet ( S^2 )
~\equil~ \{ f ( \Omega ) \oplus g ( \Omega ) \in \IC [ \Omega ]~
\vert~ f  ( 0 ) = g ( 0 ) \}}
%
where $\IC [ \Omega ]$ is the polynomial algebra in a variable
$\Omega$ of degree $2$.

In terms of the Cartan model we begin by writing the Cartan differential as
$d_\cC = d - \O~ \iota_{\p / \p \phi}$.
Then $h_1( \O ) ( \alpha + \O \cos\theta) + h_2(\O)$ is a cohomology class, 
where $\alpha$ is the solid angle, in local coordinates: $d \phi~ d(\cos\theta)$. 
Remembering that $\O$ has degree 2, we recover the description 
\ECSoSt\  of the cohomology with 
$f(\O)=h_2(\Omega) $ and $g(\O)=h_2(0)+\O h_1(\O)$. 


\subsec{Example 2: $\CG$-Equivariant cohomology of $\CA$}

We will discuss this in much greater detail in chapter \sTYMT, but for the present
we note that the space of gauge connections, $\CA$, 
on a principal bundle is the universal bundle 
for $\CG$, the group of gauge transformations. 

The Cartan model can be represented as
%
\eqn\dondii{\eqalign{
d_\cC A &=\psi\cr
d_\cC \psi & = - D_A \phi \cr
d_\cC \phi  &=0 \cr}}
%
where $\psi= \widetilde d A$.
In less condensed notation, $A_\mu^a(x)$ are ``coordinates" on $\CA$; 
$\psi_\mu^a(x) = \widetilde d A_\mu^a(x)$ are a basis of 1-forms; and $\phi^a(x)$
are functions on the Lie algebra.

\exercise{}

a.) Prove the second equation in \dondii. 

b.) Show that,
%
\eqn\rtwoclss{\eqalign{
\CO_2^{(0)}(P) & = {1\over 8 \pi^2} \Tr \phi^2(P)\cr
\CO_2^{(1)}(\gamma) & = {1\over 4 \pi^2} 
\int_\gamma \Tr(\phi \psi)\cr
\CO_2^{(2)}(\Sigma) & = {1\over 4 \pi^2} 
\int_\Sigma \Tr(\phi F- \half \psi\wedge \psi)\cr}}
%
are closed equivariant forms on $\CA$.
Here $P$ is a point in $X$, $\gamma$ is a curve and $\Sigma$ is a surface.
We will return to these forms in several later sections. 

\endexercise


\subsec{Equivariant Cohomology vs. Lie-Algebra Cohomology}
\subseclab\ssECvLAC

In the BRST quantization of gauge theories one works with Lie-algebra cohomology;
in topological field theory one works with equivariant cohomology.
It is natural to wonder how these two cohomologies are related to one another. 
It turns out that\foot{We would like to thank G. Zuckerman for explaining 
this to us.} {\it equivariant cohomology of a Lie algebra $\lieg$ is the same as a ``supersymmetrized" Lie algebra cohomology of a corresponding graded Lie algebra
$\lieg_{\rm super}$}. 

\subsubsec{Lie Algebra cohomology and BRST quantization}

Let $\lieg$ be the Lie algebra spanned by $T_i$ with $[T_i, T_j] = f^k{}_{ij} T_k$.
The ordinary Lie algebra cohomology is defined using the complex
$\Lambda^\bullet \lieg\dual$. 
We can take generators to be anticommuting elements $c^i$ of degree 1.
The action of the differential is: 
%
\eqn\brstop{
Q_{\lieg} c^i = -\half f^i{}_{jk} c^j c^k}
%
$Q_\lieg$ squares to zero by the Jacobi identity. 
Introducing a conjugate operator
$$
\{ b_i, c^j\} = \delta_i^j
$$
we can write $Q_\lieg=-\half f^i{}_{jk} c^j c^kb_i$. 
The cohomology of $Q_\lieg$, $H_Q^\bullet (\lieg)$, is the {\it Lie-algebra cohomology}. 

Now let $V$ be a $\lieg$-module.
We can define the Lie-algebra cohomology with coefficients in $V$,
$H_{Q_\lieg}^\bullet (\lieg,V)$,  by considering the complex
%
\eqn\brstcplx{
\Lambda^\bullet \lieg\dual \to \Lambda^\bullet \lieg\dual~\otimes V}
%
and the action of the differential
%
\eqn\brstdffl{
Q_{\lieg} \to  c^i \otimes \rho(T_i) + Q_{\lieg}}
%
where $\rho ( T_i )$ is the representation of $T_i$ in $V$.
In more ``physical " notation we have:
%
\eqn\brstdffi{
Q_\lieg = c^i \rho(T_i) - \half f^i_{~jk} c^j c^k b_i}
%
\par\noindent
{\bf Example:} One famous example is string theory where $\lieg=Vir(c)$ is the Virasoro
algebra, where 
$V$ is a representation provided by a CFT of central charge $c$. 
It is a well-known fact that $Q_\lieg$ only squares to zero for $c=26$.
In this case (as in the ordinary one) the cohomology defines the space of physical states. 

Note that all the above goes through if we replace a Lie algebra by a super-Lie algebra.
%In this case t
The ghosts $c^i$ carry opposite statistics to the generators $T_i$. 

\subsubsec{Supersymmetrized Lie Algebra cohomology}

To the Lie algebra $\lieg$ we now associate a differential graded Lie algebra (DGLA)
$\lieg[\epsilon]= \lieg \otimes \Lambda^\ast  \epsilon$ where $\deg( \epsilon )=-1$
and $\deg( T_i )=0$;  $\epsilon$ is Grassmann, so $\epsilon^2=0$.
The differential is defined by $\p \epsilon = 1$.  
The DGLA, $\lieg[ \epsilon ]$, is generated by $T_i$ and $\widetilde T_i =T_i \otimes \epsilon$.
It has structure constants: 
%
\eqn\dglast{
\eqalign{
[ T_i, T_j] &= f^k{}_{ij} T_k\cr
[ T_i, \widetilde T_j ]&=f^k{}_{ij} \widetilde T_k\cr
[ \widetilde T_i, \widetilde T_j ]&=0\cr}}
%
The graded exterior algebra,  $\Lambda^\bullet \lieg[\epsilon]\dual$, may be identified
with $S( \lieg\dual~ ) \otimes \Lambda( \lieg\dual~ )$. 
Indeed it is generated by $\gamma^i$ and $c^i$ of degrees 2 and 1, respectively.
The differential is defined by:
%
\eqn\dgondf{
\p\dual~ c^i = \gamma^i \qquad \p~\dual~ \gamma^i =0}
%
where $\p\dual$ is dual to $\p$. 

This super-Lie algebra has a BRST differential for $\lieg[\epsilon]$-Lie-algebra cohomology.
We introduce $b_i$ and $\beta_i$ in the usual way and get the differentials: 
%
\eqn\suprlacoh{\eqalign{
Q_{\lieg[\epsilon]} &=-f^i{}_{jk}  c^j \gamma^k \beta_i-
\half f^i{}_{jk} c^j c^k b_i  \cr
\p\dual & = \gamma^i b_i \cr
d_\CW & = Q_{\lieg[\epsilon]} + \p\dual .\cr}}
% 
where the last line makes use of the remark at the end of 
section \ssWA. 

Moreover, let $V$ be a $\lieg$-module.
Then we can promote $V\to \O^\bullet(V)$  to get a $\lieg[\epsilon]$-module
with $X$ and $\widetilde X$ acting by
%
\eqn\dgmi{
X\to \CL_X \qquad \qquad X\otimes \epsilon \to \iota_X . }
%
In fact, $( \O^\bullet ( V ), d )$ is a differential graded module (DGM) for the DGLA
$\lieg[\epsilon]$. 
The total differential: 
%
\eqn\ttldiff{
Q  =c^i \CL_i + \gamma^i \iota_i + Q_{\lieg[\epsilon]}
 +  \p\dual \otimes 1+1\otimes d}
%
coincides with the ``BRST" model differential of section \sssBRSTM ! 

\subsec{Equivariant Cohomology and Twisted $N=2$ Supersymmetry}
\subseclab\ssECandTNTwoS

Twisted $N=2$ supersymmetry algebras are closely related to equivariant cohomology. 
For example, consider the twisted $N=2$ supersymmetry algebra in two dimensions.
The relations \foot{Note that in this context $L_0$ denotes the zero mode of the
bosonic stress-energy tensor.}: 
%
\eqn\twsttii{\eqalign{
[J_0, G_0]=-G_0 \qquad & \qquad [ {\cal J}, \iota_\xi ] = - \iota_\xi\cr
[J_0, Q_0]=+ Q_0  \qquad & \qquad [ {\cal J}, d ] = d \cr
[G_0, Q_0]=L_0  \qquad & \qquad [ d, \iota_\xi ] = {\cal L}_\xi\cr}}
%
show a perfect parallel with equivariant cohomology for an action generated by a single
vector field $\xi$; that is, for  $S^1$-equivariant cohomology.  
Here ${\cal J}$ measures the degree of the form. 

There is a beautiful generalization of this \refs{\segallct,\storai}. 
Consider the loop space of a manifold $X$, which we will denote $LX$. 
Consider differential forms on $LX$, denoted $\O^\bullet (LX)$.
Formally we may think of $\O^\bullet ( LX )$ as a continuous tensor 
product of forms on $X$: 
$\O^\bullet ( LX ) = \otimes_{\theta\in S^1} \O^\bullet ( X )_\theta$.
{}From this point of view we may speak of 
a ``local degree" and a ``local exterior derivative". 
If $f(\theta)$ is a function on the circle we may form
%
\eqn\geomntwo{\eqalign{
\deg_f \equiv {\cal J}_f  & \equiv \oint d \theta~ f ( \theta )~ \widetilde d X^\mu ( \theta )~
\iota( {\p \over \p X^\mu(\theta)} )\cr
d_f = \oint f(\theta)  d_\theta & = \oint d \theta~ f(\theta)~ \widetilde d X^\mu(\theta)~
{\delta \over \delta X^\mu(\theta)}\cr}}
%

On the other hand, there is a natural action of $G= \Diff(S^1)$ on
$LX$, and given an element of the Lie algebra,  $v(\theta){\p \over \p \theta}$, there
is a corresponding vector field 
$$V=
\oint v ( \theta ) {{\partial X^\mu ( \theta )}\over{\partial \theta}} {\p \over \p X^\mu(\theta)}
$$
on $LX$.
Accordingly we have operators $L_V$ and $\iota_V$ on $\O^\bullet (LX)$. 
One easily checks that these operators 
satisfy the algebra: 
%
\eqn\gmntwii{\eqalign{
[ L_{V_1}, L_{V_2}] = L_{[ V_1, V_2]}\qquad &\qquad  [ {\cal J}_f, {\cal J}_g] = 0\cr
[ L_{V_1}, \iota_{V_2}] = \iota_{[ V_1,V_2]} \qquad &\qquad [ {\cal J}_f, \iota_V] = -\iota_{f V}\cr
[ L_{V},d_f] = -d_{f^\prime V } \qquad &\qquad  [ {\cal J}_f, d_g] = d_{f g} \cr
[\iota_V, d_f]_+ = L_{f V} & - {\cal J}_{V f'} \cr
[L_V, {\cal J}_f] &= - {\cal J}_{V f'} \cr}}
%
Defining Fourier modes via $f_n=e^{i n \theta}, v_n = e^{i n \theta}$, we have: 
%
\eqn\ntwoalg{\eqalign{
{\cal J}_{f_n}  & \to {\cal J}_n\cr
d_{f_n} & \to G_n^+\cr
\iota_{V_n}  & \to -i G_n^-\cr
L_{V_n} & \to -i L_n\cr}}
%
in which case \gmntwii\ becomes the topologically twisted $N=2$ superconformal algebra with 
central extension $c=0$.
Recall that the twisted $N=2$ algebra with central charge $c$ is generated by
$L_m,J_m,G_m$ and $Q_m$ with $m\in \IZ$ and relations: 
\eqn\twstntwo{
\eqalign{
[L_m, L_n ] = (m-n)L_{m+n} \qquad &\qquad  [J_m, J_n] = 0\cr
[L_m, G_n] =(m-n)G_{m+n}\qquad &\qquad [J_m, G_n] = -G_{m+n} \cr
[L_m,Q_n] = -n Q_{m+m} \qquad &\qquad  [J_m, Q_n] = Q_{m+n} \cr
[G_m,Q_n]_+ = 2 L_{m+n} & +n J_{m+n}+\half c m(m+1)\delta_{m+n,0} \cr
[L_m,J_n] &= - nJ_{m+n}-\half c m(m+1) \delta_{m+n,0} \cr}}
%
See, for example, \refs{\lvw, \DiVeVe}.
Putting $f=1$ in \gmntwii\ we see that ${\cal J}_f$, $d_f$, $\iota_v$, and $\CL_v$ satisfy 
the basic relations needed to define $\Diff(S^1)$-equivariant cohomology. 

\subsec{Equivariant Cohomology and Symplectic Group Actions}
\subseclab\ssECSGA

Equivariant cohomology finds a very natural application in the important example of
Hamiltonian actions of Lie groups $G$ on symplectic manifolds $( M, \omega)$, 
where $\omega= \ha \omega_{ij} dx^i dx^j$ is the 
symplectic form. In this case 
there are vector fields $V_a$ acting on $M$:
%
\eqn\algvees{
[ V_a, V_b] = f_{~ab}^c V_c}
%
with corresponding Hamiltonians generating the flows: 
%
\eqn\hamilts{
\iota_{V_a} \omega = - dH_a . }
% 
Equation \hamilts\ 
 has a lovely reinterpretation in 
equivariant cohomology.  Note that  the 
Cartan differential becomes 
\eqn\sympdffl{
D= d - \phi^a \iota_{V_a}. 
}
The definition \hamilts\ is equivalent to the statement 
 that $\omega-\phi^a H_a$
is an equivariantly closed form: 
\eqn\evclsd{
D(\omega-\phi^a H_a) = 0 . 
}
This will be useful in our discussion of localization
below.

\subsubsec{Case of \ymt}
\subsubseclab\CofYMT

This remark becomes particularly interesting in 
the context of \ymt. 
As we saw in subsection \sssetzfc, $\CA$ carries a 
natural symplectic structure in $D=2$. 
Here the moment map for the action of 
the group of gauge transformations is
simply \AtBoym:
$$\mu(A) = -{1\over 4 \pi^2} F $$
so, from \rtwoclss\ we recognize $\CO_2^{(2)}$ as 
the equivariant extension of the symplectic form. 
\def\Ahl{[\hyperref {}{reference}{1}{1}]}
\def\Al{[\hyperref {}{reference}{2}{2}]}
\def\AGIndex{[\hyperref {}{reference}{3}{3}]}
\def\lagpg{[\hyperref {}{reference}{4}{4}]}
\def\AnGaNaTa{[\hyperref {}{reference}{5}{5}]}
\def\ArCo{[\hyperref {}{reference}{6}{6}]}
\def\Ar{[\hyperref {}{reference}{7}{7}]}
\def\AsMo{[\hyperref {}{reference}{8}{8}]}
\def\atising{[\hyperref {}{reference}{9}{9}]}
\def\AHS{[\hyperref {}{reference}{10}{10}]}
\def\AtBoym{[\hyperref {}{reference}{11}{11}]}
\def\AtBomm{[\hyperref {}{reference}{12}{12}]}
\def\atiyahsinger{[\hyperref {}{reference}{13}{13}]}
\def\atiythrfr{[\hyperref {}{reference}{14}{14}]}
\def\atiyax{[\hyperref {}{reference}{15}{15}]}
\def\AtJe{[\hyperref {}{reference}{16}{16}]}
\def\BaVi{[\hyperref {}{reference}{17}{17}]}
\def\bazrev{[\hyperref {}{reference}{18}{18}]}
\def\BaTa{[\hyperref {}{reference}{19}{19}]}
\def\demeterfi{[\hyperref {}{reference}{20}{20}]}
\def\Ba{[\hyperref {}{reference}{21}{21}]}
\def\BaHa{[\hyperref {}{reference}{22}{22}]}
\def\BaSiii{[\hyperref {}{reference}{23}{23}]}
\def\BaSi{[\hyperref {}{reference}{24}{24}]}
\def\BaSii{[\hyperref {}{reference}{25}{25}]}
\def\BaTh{[\hyperref {}{reference}{26}{26}]}
\def\becchi{[\hyperref {}{reference}{27}{27}]}
\def\BeGi{[\hyperref {}{reference}{28}{28}]}
\def\bgv{[\hyperref {}{reference}{29}{29}]}
\def\bers{[\hyperref {}{reference}{30}{30}]}
\def\BeCeOoVa{[\hyperref {}{reference}{31}{31}]}
\def\BeEd{[\hyperref {}{reference}{32}{32}]}
\def\Bir{[\hyperref {}{reference}{33}{33}]}
\def\bbrt{[\hyperref {}{reference}{34}{34}]}
\def\Bis{[\hyperref {}{reference}{35}{35}]}
\def\Bl{[\hyperref {}{reference}{36}{36}]}
\def\BlThlgt{[\hyperref {}{reference}{37}{37}]}
\def\BlThqym{[\hyperref {}{reference}{38}{38}]}
\def\blauthom{[\hyperref {}{reference}{39}{39}]}
\def\BogSh{[\hyperref {}{reference}{40}{40}]}
\def\Bo{[\hyperref {}{reference}{41}{41}]}
\def\BoTu{[\hyperref {}{reference}{42}{42}]}
\def\BoHuMaMi{[\hyperref {}{reference}{43}{43}]}
\def\Braam{[\hyperref {}{reference}{44}{44}]}
\def\Br{[\hyperref {}{reference}{45}{45}]}
\def\BRSTrefs{[\hyperref {}{reference}{46}{46}]}
\def\BrSchSch{[\hyperref {}{reference}{47}{47}]}
\def\CaDeGrPa{[\hyperref {}{reference}{48}{48}]}
\def\ccd{[\hyperref {}{reference}{49}{49}]}
\def\chen{[\hyperref {}{reference}{50}{50}]}
\def\Co{[\hyperref {}{reference}{51}{51}]}
\def\CMROLD{[\hyperref {}{reference}{52}{52}]}
\def\CMRPI{[\hyperref {}{reference}{53}{53}]}
\def\CMRPII{[\hyperref {}{reference}{54}{54}]}
\def\CMRIII{[\hyperref {}{reference}{55}{55}]}
\def\CreTay{[\hyperref {}{reference}{56}{56}]}
\def\danvia{[\hyperref {}{reference}{57}{57}]}
\def\wadia{[\hyperref {}{reference}{58}{58}]}
\def\DiInMoPl{[\hyperref {}{reference}{59}{59}]}
\def\DiRu{[\hyperref {}{reference}{60}{60}]}
\def\DiVVTrieste{[\hyperref {}{reference}{61}{61}]}
\def\DiWi{[\hyperref {}{reference}{62}{62}]}
\def\Di{[\hyperref {}{reference}{63}{63}]}
\def\DiSeWeWi{[\hyperref {}{reference}{64}{64}]}
\def\DoApp{[\hyperref {}{reference}{65}{65}]}
\def\Don{[\hyperref {}{reference}{66}{66}]}
\def\DoKro{[\hyperref {}{reference}{67}{67}]}
\def\dgcrg{[\hyperref {}{reference}{68}{68}]}
\def\Dosc{[\hyperref {}{reference}{69}{69}]}
\def\Docft{[\hyperref {}{reference}{70}{70}]}
\def\DGLSS{[\hyperref {}{reference}{71}{71}]}
\def\DoKa{[\hyperref {}{reference}{72}{72}]}
\def\DrZu{[\hyperref {}{reference}{73}{73}]}
\def\Du{[\hyperref {}{reference}{74}{74}]}
\def\Ed{[\hyperref {}{reference}{75}{75}]}
\def\egh{[\hyperref {}{reference}{76}{76}]}
\def\echikann{[\hyperref {}{reference}{77}{77}]}
\def\Ez{[\hyperref {}{reference}{78}{78}]}
\def\Fi{[\hyperref {}{reference}{79}{79}]}
\def\FlSympi{[\hyperref {}{reference}{80}{80}]}
\def\FlSympii{[\hyperref {}{reference}{81}{81}]}
\def\FlWitt{[\hyperref {}{reference}{82}{82}]}
\def\FlThreeD{[\hyperref {}{reference}{83}{83}]}
\def\Fo{[\hyperref {}{reference}{84}{84}]}
\def\Freed{[\hyperref {}{reference}{85}{85}]}
\def\FrieWind{[\hyperref {}{reference}{86}{86}]}
\def\Fu{[\hyperref {}{reference}{87}{87}]}
\def\ganor{[\hyperref {}{reference}{88}{88}]}
\def\getzler{[\hyperref {}{reference}{89}{89}]}
\def\GlJa{[\hyperref {}{reference}{90}{90}]}
\def\Grtalk{[\hyperref {}{reference}{91}{91}]}
\def\GrHa{[\hyperref {}{reference}{92}{92}]}
\def\Gr{[\hyperref {}{reference}{93}{93}]}
\def\GrTa{[\hyperref {}{reference}{94}{94}]}
\def\grssmatyt{[\hyperref {}{reference}{95}{95}]}
\def\GrMat{[\hyperref {}{reference}{96}{96}]}
\def\GrTatalk{[\hyperref {}{reference}{97}{97}]}
\def\GrWi{[\hyperref {}{reference}{98}{98}]}
\def\GrKiSe{[\hyperref {}{reference}{99}{99}]}
\def\hamermesh{[\hyperref {}{reference}{100}{100}]}
\def\Guest{[\hyperref {}{reference}{101}{101}]}
\def\harstrom{[\hyperref {}{reference}{102}{102}]}
\def\HaMo{[\hyperref {}{reference}{103}{103}]}
\def\HaMu{[\hyperref {}{reference}{104}{104}]}
\def\Hi{[\hyperref {}{reference}{105}{105}]}
\def\Hitchin{[\hyperref {}{reference}{106}{106}]}
\def\Hora{[\hyperref {}{reference}{107}{107}]}
\def\Hott{[\hyperref {}{reference}{108}{108}]}
\def\Hori{[\hyperref {}{reference}{109}{109}]}
\def\Hurt{[\hyperref {}{reference}{110}{110}]}
\def\husemoller{[\hyperref {}{reference}{111}{111}]}
\def\Ing{[\hyperref {}{reference}{112}{112}]}
\def\isham{[\hyperref {}{reference}{113}{113}]}
\def\Kal{[\hyperref {}{reference}{114}{114}]}
\def\Kan{[\hyperref {}{reference}{115}{115}]}
\def\KaNi{[\hyperref {}{reference}{116}{116}]}
\def\KaKo{[\hyperref {}{reference}{117}{117}]}
\def\Kazkos{[\hyperref {}{reference}{118}{118}]}
\def\Kazwynt{[\hyperref {}{reference}{119}{119}]}
\def\kazrev{[\hyperref {}{reference}{120}{120}]}
\def\Ki{[\hyperref {}{reference}{121}{121}]}
\def\Kirwan{[\hyperref {}{reference}{122}{122}]}
\def\jeffkir{[\hyperref {}{reference}{123}{123}]}
\def\Kn{[\hyperref {}{reference}{124}{124}]}
\def\kobnom{[\hyperref {}{reference}{125}{125}]}
\def\Kon{[\hyperref {}{reference}{126}{126}]}
\def\kontsevichi{[\hyperref {}{reference}{127}{127}]}
\def\kontsevichii{[\hyperref {}{reference}{128}{128}]}
\def\Kos{[\hyperref {}{reference}{129}{129}]}
\def\kronmrow{[\hyperref {}{reference}{130}{130}]}
\def\Ku{[\hyperref {}{reference}{131}{131}]}
\def\LaPeWi{[\hyperref {}{reference}{132}{132}]}
\def\lvw{[\hyperref {}{reference}{133}{133}]}
\def\lz{[\hyperref {}{reference}{134}{134}]}
\def\lossev{[\hyperref {}{reference}{135}{135}]}
\def\manin{[\hyperref {}{reference}{136}{136}]}
\def\Ma{[\hyperref {}{reference}{137}{137}]}
\def\MaQu{[\hyperref {}{reference}{138}{138}]}
\def\MDSa{[\hyperref {}{reference}{139}{139}]}
\def\Mig{[\hyperref {}{reference}{140}{140}]}
\def\MitH{[\hyperref {}{reference}{141}{141}]}
\def\Mij{[\hyperref {}{reference}{142}{142}]}
\def\miller{[\hyperref {}{reference}{143}{143}]}
\def\milnor{[\hyperref {}{reference}{144}{144}]}
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%
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%hepth 9407176. }
%
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\ifstudvers
  \def\remove{\begingroup\obeylines\newlinechar=`\^^M\stripline}
  {\obeylines\gdef\stripline#1^^M{\toks0{#1}\edef\next{\the\toks0}%
  \ifx\next\eMark\let\next=\endgroup\else\let\next=\stripline\fi\next}}
  \def\eMark{\endremove}
  \def\endremove{\empty}
  \def\nl{\par\nobreak}
\else
  \def\remove{\vskip0.1truein\hrule height 2pt\vskip0.1truein}
  \def\endremove{\vskip0.1truein\hrule height 2pt\vskip0.1truein}
\fi

% Something to deal with sub-sub-sections
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi  %if desired
\global\subsubsecno=0%
\global\subsecno=0\eqnres@t\let\s@csym\secsym\xdef\secn@m{\the\secno}\noindent
{\bf\hyperdef\hypernoname{section}{\the\secno}{\the\secno.} #1}%
\writetoca{{\string\hyperref{}{section}{\the\secno}{\the\secno.}} {#1}}%
\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1\message{(\s@csym\the\subsecno. #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\global\subsubsecno=0%
\noindent{{\it\hyperdef\hypernoname{subsection}{\secn@m.\the\subsecno}%
{\secn@m.\the\subsecno.}} #1}\writetoca{\string\quad
{\string\hyperref{}{subsection}{\secn@m.\the\subsecno}{\secn@m.\the\subsecno.}}
{#1}}\par\nobreak\medskip\nobreak}
%
\global\newcount\subsubsecno \global\subsubsecno=0
\def\subsubsec#1{\global\advance\subsubsecno
by1\message{(\s@csym\the\subsecno.\the\subsubsecno. #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\noindent\quad{\it\hyperdef\hypernoname{subsubsection}%
{\secn@m.\the\subsecno.\the\subsubsecno}%
{\secn@m.\the\subsecno.\the\subsubsecno.} {#1}}\writetoca{\string\qquad 
{\string\hyperref{}{subsubsection}{\secn@m.\the\subsecno.\the\subsubsecno}
{\secn@m.\the\subsecno.\the\subsubsecno.}} {#1}}%
\par\nobreak\medskip\nobreak}

\def\subsubseclab#1{\DefWarn#1\xdef #1{\noexpand\hyperref{}{subsubsection}%
{\secn@m.\the\subsecno.\the\subsubsecno}%
{\secn@m.\the\subsecno.\the\subsubsecno}}%
\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}

% Macros for boxes

\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern8pt
\vbox{\hbox{\kern8pt}\hbox{\vbox{#1}}\hbox{\kern8pt}}
\kern8pt\vrule}\hrule}}
\def\mathboxit#1{\vbox{\hrule\hbox{\vrule\kern8pt\vbox{\kern8pt
\hbox{$\displaystyle #1$}\kern8pt}\kern8pt\vrule}\hrule}}

% Other macros

% for finite dimensional Lie algebra:
% for Lie algebra of group of gauge transformations:

\def\ab {\bar{\alpha}}
\def\Ad{{\mathop{\rm Ad}}}
\def\Aut{{\mathop {\rm Aut}}}
\def\bA{{\bf A}}
\def\bb {\bar{b}}
\def\bC{{\IC}}
\def\bD{{\ID}}
\def\bF{{\IF}}
\def\bG{{\IG}}
\def\bFm{\pmatrix{{\widetilde f}_{\gamma i}\cr
                  {\widetilde g}^\gamma\cr}}
\def\bN{{\IN}}
\def\bO{{\IO}}
\def\bOdef{\pmatrix{
\delta^i_{~ j} \nabla_\alpha + J^i_{~ j} \epsilon_\alpha^{~ \beta} \nabla_\beta&
-J^i_{~ j} \partial^\gamma f^j \epsilon_\alpha^{~ \beta}\cr
h^{\alpha\beta} \partial_\beta f^i G_{ij}&
- h^{\alpha\beta} D^\gamma\cr}}
\def\bP{{\IP}}
\def\bPi{{\IPi}}
\def\bPim{\pmatrix{{\widetilde \pi}^i\cr
{\widetilde\pi}_{\alpha\beta}\cr}}
\def\bPimt{\pmatrix{{\widetilde \pi}_i &
{\widetilde\pi}_{\alpha\beta}\cr}}
\def\bQ{{\bf Q}}
\def\bR{{\IR}}
\def\bRm{\pmatrix{\widetilde\rho_{\gamma i}\cr
\widetilde\sigma^\gamma\cr}}
\def\bV{{\bf V}}
\def\bX{{\bf X}}
\def\bXm{\pmatrix{{\widetilde\chi}^i \cr
{\widetilde\psi}_{\alpha\beta}\cr}}
\def\bXmt{\pmatrix{{\widetilde\chi}^i &
{\widetilde\psi}_{\alpha\beta}\cr}}
\def\bZ{{\bf Z}}
\def\c{\cdot}
\def\cA{{\cal A}}
\def\cb {\bar{c}}
\def\cC{{\cal C}}
\def\cch{ {\cal \chi} }
\def\cD{{\cal D}}
\def\CD {{\cal D}}
\def\cE{{\cal E}}
\def\CE {{\cal E}}
\def\cF{{\cal F}}
\def\CF {{\cal F}}
\def\cFpm{{\cF^\pm}}
\def\cG{{\cal G}}
\def\cgp {c_\Gamma^l}
\def\cgm {c_\Gamma^r }
\def\cH{{\cal H}}
\def\CI  {{\cal I}}
\def\cL{{\cal L}}
\def\CL {{\cal L}}
\def\cM{{\cal M}}
\def\CM {{\cal M}}
\def\CN {{\cal N}}
\def\cO{{\cal O}}
\def\CO {{\cal O}}
\def\codim{{\mathop{\rm codim}}}
\def\cok{{\rm cok}}
\def\coker{{\mathop {\rm coker}}}
\def\cP{{\cal P}}
\def\CP {{\cal P }}
\def\CQ {{\cal Q }}
\def\CR {{\cal R}}
\def\CW{{\cal W}}
\def\CY{{\cal Y}}
\def\cS{{\cal S}}
\def\CS {{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
\def\CV {{\cal V}}
\def\cV{{\cal V}}
\def\cVfh{{\cV_{\scriptscriptstyle (f,h)}}}
\def\cVz{{\cV_{\scriptscriptstyle 0}}}
\def\cVzfh{{\cV_{\scriptscriptstyle 0~ (f,h)}}}
\def\cW{{\cal W}}
\def\cWfh{{\cW_{\scriptscriptstyle (f,h)}}}
\def\cWz{{\cW_{\scriptscriptstyle 0}}}
\def\cWzfh{{\cW_{\scriptscriptstyle 0~ (f,h)}}}
\def\cZ {{\cal Z}}
\def\CZ {{\cal Z}}
\def\deg{{\mathop{\rm deg}}}
\def\diagonal{{\mathop{\rm Diag}}}
\def\diff{{\rm diff}}
\def\Diff{{\rm Diff}}
\def\dim{{\mathop{\rm dim}}}
\def\dual{{{}^\ast}}
\def\End{{\mathop{\rm End}}}
\def\eqdef{{\buildrel{\rm def}\over =}}
\def\equil{{\buildrel\sim\over=}}
\def\etwo { {e^2 \over 2} } 
\def\fbproj{{\buildrel{\pi}\over\longrightarrow}}
\def\FRAME{{\rm FRAME}}
\def\G {\Gamma}
\def\O{\Omega}
\def\Os{\Omega^*}
\def\hb {\hbar}
\def\half{{\textstyle{1\over 2}}}
\def\hol{{\HOL}}
\def\HOL{{\cal H}}
\def\holo{holomorphic}
\def\Holo{Holomorphic}
\def\Hom{{\mathop {\rm Hom}}}
\def\hs{{\hat s}}
\def\hS{{\hat S}}
\def \ib{{\bar i}}
\def\IB{\relax{\rm I\kern-.18em B}}
\def\IC{\relax\hbox{$\inbar\kern-.3em{\rm C}$}}
\def\ID{\relax{\rm I\kern-.18em D}}
\def\IE{\relax{\rm I\kern-.18em E}}
\def\IF{\relax{\rm I\kern-.18em F}}
\def\IG{\relax\hbox{$\inbar\kern-.3em{\rm G}$}}
\def\IGa{\relax\hbox{${\rm I}\kern-.18em\Gamma$}}
\def\IH{\relax{\rm I\kern-.18em H}}
\def\II{\relax{\rm I\kern-.18em I}}
\def\IK{\relax{\rm I\kern-.18em K}}
\def\IL{\relax{\rm I\kern-.18em L}}
\def\IM{\relax{\rm I\kern-.18em M}}
\def\Im{{\mathop{\rm Im}}}
\def\ker{{\mathop{\rm ker}}}
\def\IN{\relax{\rm I\kern-.18em N}}
\def\inclusionmap#1{{\smash{
        \mathop{\hookrightarrow}\limits^{#1}}}}
\def\ind{{\mathop{\rm ind}}}
\def\Index{{\mathop{\rm ind}}}
\def\IO{\relax\hbox{$\inbar\kern-.3em{\rm O}$}}
\def\Iom{{\inbar\kern-3.00pt\Omega}}
\def\IOm{\relax\hbox{$\inbar\kern-3.00pt\Omega$}}
\def\ipK#1#2{{\bigl ( #1, #2 \bigr )}}
\def\ipM#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\sst \cM}}}
\def\ipTM#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\sst {T \cM}}}}
\def\iptV#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\sst \tcV}}}
\def\ipV#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\sst \cV}}}
\def\ipW#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\sst \cW}}}
\def\IP{\relax{\rm I\kern-.18em P}}
\def\IPi{\relax\hbox{${\rm I}\kern-.18em\Pi$}}
\def\ipK#1#2{{\bigl ( #1, #2 \bigr )}}
\def\ipTF#1#2{{\bigl\langle #1, #2 \bigr\rangle_{T \cF}}}
\def\ipTtF#1#2{{\bigl\langle #1, #2 \bigr\rangle_{T \tcF}}}
\def\ipTtFpm#1#2{{\bigl\langle #1, #2 \bigr\rangle_{T \tcF^\pm}}}
\def\ipTM#1#2{{\bigl\langle #1, #2 \bigr\rangle_{T \cM_{-1}}}}
\def\ipTtM#1#2{{\bigl\langle #1, #2 \bigr\rangle_{T \tcM_{-1}}}}
\def\iptV#1#2{{\bigl\langle #1, #2 \bigr\rangle_\tcV}}
\def\iptncV#1#2{{\bigl\langle #1, #2 \bigr\rangle_{\tcV^{nc}}}}
\def\ipV#1#2{{\bigl\langle #1, #2 \bigr\rangle_\cV}}
\def\IQ{\relax\hbox{$\inbar\kern-.3em{\rm Q}$}}
\def\IR{\relax{\rm I\kern-.18em R}}
\def\isomorphic{{\equiv}}
\def\ITh{\relax\hbox{$\inbar\kern-.3em\Theta$}}
\def\inbar{\,\vrule height1.5ex width.4pt depth0pt}
\def\iW{{i_{\scriptscriptstyle{\cal W}}}}
\font\cmss=cmss10 \font\cmsss=cmss10 at 7pt
\def\IZ{\relax\ifmmode\mathchoice
{\hbox{\cmss Z\kern-.4em Z}}{\hbox{\cmss Z\kern-.4em Z}}
{\lower.9pt\hbox{\cmsss Z\kern-.4em Z}}
{\lower1.2pt\hbox{\cmsss Z\kern-.4em Z}}\else{\cmss Z\kern-.4em
Z}\fi}
\def \jb{{\bar j}}
\def\ker{{\mathop{\rm ker}}}
\def\kG {\vec{k}_\Gamma}
\def\liebg{{{\rm Lie}~ \CG}}
\def\lieg{{\underline{\bf g}}}
\def\lieG{\got{G}}
\def\lieh{\got{h}}
\def\lieH{\got{H}}
%\def\ll{L.L}
\def\loclor{{\rm local\ lorentz}}
\def\locLor{{\rm local\ Lorentz}}
\def\log {{\rm log}}
\def\MAP{{\rm MAP}}
\def\mapdown#1{\Big\downarrow
        \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapleft#1{\smash{
        \mathop{\longleftarrow}\limits^{#1}}}
\def\mapne#1{\nearrow
        \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapright#1{\smash{
        \mathop{\longrightarrow}\limits^{#1}}}
\def\mapse#1{\searrow
        \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapsw#1{\swarrow
        \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow
        \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\MET{{\rm MET}}
\def\Met{{\rm MET}}
\def\min{{\mathop{\rm min}}}
\def\Odagger{
\pmatrix{- ( \delta_\mu{}^\nu \nabla_\gamma + J_\mu{}^\nu \nabla_\beta \epsilon^\beta{}_\gamma ) &
- G_{\mu\nu} \partial_\gamma f^\nu\cr
- \delta^\alpha{}_\gamma J^\nu{}_\mu \partial_\delta f^\mu \epsilon^{\delta\beta} &
{1\over2} ( \delta_\gamma{}^\alpha D^\beta + \delta_\gamma{}^\beta D^\alpha ) \cr}}
%- ( \delta_\gamma{}^\alpha \delta^\mu{}_\nu
% + \epsilon_\gamma{}^\alpha J^\mu{}_\nu ) \nabla_\alpha &
%- \half ( h_{\alpha\gamma} \partial_\beta
%            + h_{\beta\gamma} \partial_\alpha ) f^\mu \cr
%- \partial_\alpha f^\nu &
%\half ( h_{\alpha\gamma} D_\beta
%               + h_{\beta\gamma} D_\alpha  )\cr}}
\def\ofp{{1\over{4\pi}}}
\def\otp{{1\over{2\pi}}}
\def\p {\partial}
\def\pb{\bar{\partial}}
\def\Pfaff{{\rm Pfaff}}
\def\Pf{{\rm Pfaff}}
\def\piV{{\pi_{\scriptscriptstyle{\cal V}}}}
\def\piW{{\pi_{\scriptscriptstyle{\cal W}}}}
\def\pmf{\IP\CM\CF}
\def\pseudo{co-}
\def\Pseudo{Co-}
\def\psm{\psi_-}
\def\psp{\psi_+}
\def\quarter{{\textstyle{1\over 4}}}
\def\Ram{{\mathop{\rm Ram}}}
\def\rank{{\mathop{\rm rank}}}
\def\Sc {\Sigma_T^c}
\def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule
height 4.1pt depth -.3pt width .25pt\hskip-2pt$\times$}}}
\def\SG{{\Sigma_T}}
\def\sh{{\sqrt{h}}}
\def\Sh{{\Sigma_W}}
\def\sign{{\mathop{\rm sign}}}
\def\sst{\scriptscriptstyle}
\def\Span{{\mathop{\rm Span}}}
\def\st{\scriptstyle}
\def\ST{{\SG}}
\def\Sw{{\Sh}}
\def\Sym{{\mathop{\rm Sym}}}
\def\tb{\bar t}
\def\tbA{{\widehat{\bA}}}
\def\tbF{{\widehat{\IF}}}
\def\tbG{{\widehat{\bG}}}
\def\tbm {{\bar{\theta}^-}} 
\def\tbp {{\bar{\theta}^+}} 
\def\tbPi{{\widehat{\IPi}}}
\def\tbR{{\widetilde{\IR}}}
\def\tbX{{\widetilde{\bf X}}}
\def\tcF{{\widetilde{\cal F}}}
\def\tcFpm{{\tcF^\pm}}
\def\tcM{{\widetilde{\cal M}}}
\def\tcN{{\widetilde{\cal N}}}
\def\tcV{{\widetilde{\cal V}}}
\def\te{\CT^{Euc}}
\def\tm {\theta^-}% Caution old conflict!!!
\def\TM{\CT^{Mink}}
\def\top{topological}
\def\Top{Topological}
\def\tp {\theta^+} 
\def\tr{{\mathop{\rm Tr}}}
\def\U {\Upsilon}
\def\vac {\vert\downarrow \downarrow,p\rangle}
\def\vol{{\rm vol}}
\def\vskipabit{{\vskip0.25truein}}
\def\wb {\bar{w}}
\def\wCO{ \hat \CO  } 
\def\weylg{{\rm weyl}}
\def\Weyl{{\rm Weyl}}
\def\ymt{$YM_2$}
\def\zb {{\bar{z}}}
%
%exercise macro stolen from PG:
%
\def\exercise#1{\bgroup\narrower\footnotefont
\baselineskip\footskip\bigbreak
\hrule\medskip\nobreak\noindent {\bf Exercise}. {\it #1\/}\par\nobreak}
\def\endexercise{\medskip\nobreak\hrule\bigbreak\egroup}
%
%
% S-Tables Macro
%
\message{S-Tables Macro v1.0, ACS, TAMU (RANHELP@VENUS.TAMU.EDU)}
%
% Help Text
%
\newhelp\stablestylehelp{You must choose a style between 0 and 3.}%
\newhelp\stablelinehelp{You should not use special hrules when
stretching
a table.}%
\newhelp\stablesmultiplehelp{You have tried to place an S-Table
inside another
S-Table.  I would recommend not going on.}%
%
% Line Thicknesses (Values)
%
\newdimen\stablesthinline
\stablesthinline=0.4pt
\newdimen\stablesthickline
\stablesthickline=1pt
%
% Border and Internal Line Thicknesses
%
\newif\ifstablesborderthin
\stablesborderthinfalse
\newif\ifstablesinternalthin
\stablesinternalthintrue
\newif\ifstablesomit
\newif\ifstablemode
\newif\ifstablesright
\stablesrightfalse
%
% Save Registers
%
\newdimen\stablesbaselineskip
\newdimen\stableslineskip
\newdimen\stableslineskiplimit
%
% Counters
%
\newcount\stablesmode
\newcount\stableslines
\newcount\stablestemp
\stablestemp=3
\newcount\stablescount
\stablescount=0
\newcount\stableslinet
\stableslinet=0
%
% Table Style Selection
%
% 0 - Centered
% 1 - Left Justified
% 2 - Right Justified
% 3 - Not Justified
%
\newcount\stablestyle
\stablestyle=0
%
% Element Buffering Definitions
%
\def\stablesleft{\quad\hfil}%
\def\stablesright{\hfil\quad}%
%
% Vertical Bar Activation
%
\catcode`\|=\active%
%
% Strut Control
%
\newcount\stablestrutsize
\newbox\stablestrutbox
\setbox\stablestrutbox=\hbox{\vrule height10pt depth5pt width0pt}
\def\stablestrut{\relax\ifmmode%
                         \copy\stablestrutbox%
                       \else%
                         \unhcopy\stablestrutbox%
                       \fi}%
%
% Misc. Internal Stuff
%
\newdimen\stablesborderwidth
\newdimen\stablesinternalwidth
\newdimen\stablesdummy
\newcount\stablesdummyc
\newif\ifstablesin
\stablesinfalse
%
% Table Macros
%
\def\begintable{\stablestart%
  \stablemodetrue%
  \stablesadj%
  \halign%
  \stablesdef}%
\def\begintableto#1{\stablestart%
  \stablemodefalse%
  \stablesadj%
  \halign to #1%
  \stablesdef}%
\def\begintablesp#1{\stablestart%
  \stablemodefalse%
  \stablesadj%
  \halign spread #1%
  \stablesdef}%
\def\stablesadj{%
  \ifcase\stablestyle%
    \hbox to \hsize\bgroup\hss\vbox\bgroup%
  \or%
    \hbox to \hsize\bgroup\vbox\bgroup%
  \or%
    \hbox to \hsize\bgroup\hss\vbox\bgroup%
  \or%
    \hbox\bgroup\vbox\bgroup%
  \else%
    \errhelp=\stablestylehelp%
    \errmessage{Invalid style selected, using default}%
    \hbox to \hsize\bgroup\hss\vbox\bgroup%
  \fi}%
\def\stablesend{\egroup%
  \ifcase\stablestyle%
    \hss\egroup%
  \or%
    \hss\egroup%
  \or%
    \egroup%
  \or%
    \egroup%
  \else%
    \hss\egroup%
  \fi}%
\def\stablestart{%
  \ifstablesin%
    \errhelp=\stablesmultiplehelp%
    \errmessage{An S-Table cannot be placed within an S-Table!}%
  \fi
  \global\stablesintrue%
  \global\advance\stablescount by 1%
  \message{<S-Tables Generating Table \number\stablescount}%
  \begingroup%
  \stablestrutsize=\ht\stablestrutbox%
  \advance\stablestrutsize by \dp\stablestrutbox%
  \ifstablesborderthin%
    \stablesborderwidth=\stablesthinline%
  \else%
    \stablesborderwidth=\stablesthickline%
  \fi%
  \ifstablesinternalthin%
    \stablesinternalwidth=\stablesthinline%
  \else%
    \stablesinternalwidth=\stablesthickline%
  \fi%
  \tabskip=0pt%
  \stablesbaselineskip=\baselineskip%
  \stableslineskip=\lineskip%
  \stableslineskiplimit=\lineskiplimit%
  \offinterlineskip%
  \def\borderrule{\vrule width \stablesborderwidth}%
  \def\internalrule{\vrule width \stablesinternalwidth}%
%
  \def\thinline{\noalign{\hrule height \stablesthinline}}%
  \def\thickline{\noalign{\hrule height \stablesthickline}}%
  \def\trule{\omit\leaders\hrule height \stablesthinline\hfill}%
  \def\ttrule{\omit\leaders\hrule height \stablesthickline\hfill}%
  \def\tttrule##1{\omit\leaders\hrule height ##1\hfill}%
  \def\stablesel{&\omit\global\stablesmode=0%
    \global\advance\stableslines by 1\borderrule\hfil\cr}%
  \def\el{\stablesel&}%
  \def\elt{\stablesel\thinline&}%
  \def\eltt{\stablesel\thickline&}%
  \def\elttt##1{\stablesel\noalign{\hrule height ##1}&}%
  \def\elspec{&\omit\hfil\borderrule\cr\omit\borderrule&%
              \ifstablemode%
              \else%
                \errhelp=\stablelinehelp%
                \errmessage{Special ruling will not display
properly}%
              \fi}%
%
  \def\stmultispan##1{\mscount=##1 \loop\ifnum\mscount>3
\stspan\repeat}%
  \def\stspan{\span\omit \advance\mscount by -1}%
%
  \def\multicolumn##1{\omit\multiply\stablestemp by ##1%
     \stmultispan{\stablestemp}%
     \advance\stablesmode by ##1%
     \advance\stablesmode by -1%
     \stablestemp=3}%

\def\multirow##1{\stablesdummyc=##1\parindent=0pt\setbox0\hbox\bgroup%

    \aftergroup\emultirow\let\temp=}
  \def\emultirow{\setbox1\vbox to\stablesdummyc\stablestrutsize%
    {\hsize\wd0\vfil\box0\vfil}%
    \ht1=\ht\stablestrutbox%
    \dp1=\dp\stablestrutbox%
    \box1}%
%
%  \def\stvcen##1{\vtop{\vfill\hbox{##1}\vfill}}% Currently does not work!
  \def\stpar##1{\vtop\bgroup\hsize ##1%
     \baselineskip=\stablesbaselineskip%
     \lineskip=\stableslineskip%

\lineskiplimit=\stableslineskiplimit\bgroup\aftergroup\estpar\let\temp
=}%
  \def\estpar{\vskip 6pt\egroup}%
  \def\stparrow##1##2{\stablesdummy=##2%
     \setbox0=\vtop to ##1\stablestrutsize\bgroup%
     \hsize\stablesdummy%
     \baselineskip=\stablesbaselineskip%
     \lineskip=\stableslineskip%
     \lineskiplimit=\stableslineskiplimit%
     \bgroup\vfil\aftergroup\estparrow%
     \let\temp=}%
  \def\estparrow{\vfil\egroup%
     \ht0=\ht\stablestrutbox%
     \dp0=\dp\stablestrutbox%
     \wd0=\stablesdummy%
     \box0}%
%
  \def|{\global\advance\stablesmode by 1&&&}%
  \def\|{\global\advance\stablesmode by 1&\omit\vrule width 0pt%
         \hfil&&}%
  \def\vt{\global\advance\stablesmode by 1&\omit\vrule width
\stablesthinline%
          \hfil&&}%
  \def\vtt{\global\advance\stablesmode by 1&\omit\vrule width
\stablesthickline%
          \hfil&&}%
  \def\vttt##1{\global\advance\stablesmode by 1&\omit\vrule width
##1%
          \hfil&&}%
  \def\vtr{\global\advance\stablesmode by 1&\omit\hfil\vrule width%
           \stablesthinline&&}%
  \def\vttr{\global\advance\stablesmode by 1&\omit\hfil\vrule width%
            \stablesthickline&&}%
  \def\vtttr##1{\global\advance\stablesmode by 1&\omit\hfil\vrule
width ##1&&}%
  \stableslines=0%
  \stablesomitfalse}
%
\def\stablesdef{\bgroup\stablestrut\borderrule##\tabskip=0pt plus
1fil%
  &\stablesleft##\stablesright%

&##\ifstablesright\hfill\fi\internalrule\ifstablesright\else\hfill\fi%

  \tabskip 0pt&&##\hfil\tabskip=0pt plus 1fil%
  &\stablesleft##\stablesright%

&##\ifstablesright\hfill\fi\internalrule\ifstablesright\else\hfill\fi%

  \tabskip=0pt\cr%
  \ifstablesborderthin%
    \thinline%
  \else%
    \thickline%
  \fi&%
}%
\def\endtable{\advance\stableslines by 1\advance\stablesmode by 1%
   \message{- Rows: \number\stableslines, Columns:
\number\stablesmode>}%
   \stablesel%
   \ifstablesborderthin%
     \thinline%
   \else%
     \thickline%
   \fi%
   \egroup\stablesend%
\endgroup%
\global\stablesinfalse}
%
% end of STABLES.TEX
%
\lockat

\input chptr1.tex
\input chptr2.tex
\input chptr3.tex
\input chptr4.tex
\input chptr5.tex
\input chptr6.tex
\input chptr7.tex
\input chptr8.tex
\input chptr9.tex
\input chptr10.tex
\input chptr11.tex
\input chptr12.tex
\input chptr13.tex
\input chptr14.tex
\input chptr15.tex
\input chptr16.tex
\input chptr17.tex
\input chptr18.tex
\input chptr19.tex
\input chptr20.tex
\input chptr21.tex


\listrefs

\bye


