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\title{{\bf The symplectic structure on moduli space}\\(in memory of
Andreas Floer)} 
\author{Alan Weinstein\thanks{Research partially supported by NSF
Grant DMS-90-01089.}
\\Department of Mathematics\\
University of California\\
Berkeley, CA 94720 USA\\
{\small(alanw@math.berkeley.edu)}}
\date{}
\begin{document}
\maketitle
%\tableofcontents
\section{Introduction}
\label{sec-intro}

The ``moduli space'' referred to in the title of this paper is the
space $\calm$ of gauge equivalence classes of flat connections on a
principal $G$ bundle over a compact Riemann surface $\Sigma $.  Atiyah
and Bott \cite{at-bo:yang-mills} constructed a symplectic structure
$\omega_{\calm } $ on $\calm $ by symplectic reduction from the
infinite dimensional symplectic manifold of {\em all} connections.
Since $\calm $ is a finite-dimensional object, it seems desirable to
have a finite-dimensional construction of its symplectic form.  In
fact, $\calm $ can also be realized as the representation space
$\hompig/G$, where $\pi$ is the fundamental group of $\Sigma $ and $G$
acts by conjugation.  Using the resulting identification of the
tangent spaces of $\calm $ with cohomology spaces of $\pi $ with
suitable coefficients, Goldman \cite{go:symplectic} gave a direct
construction of $\omega_{\calm } $ as a nondegenerate 2-form, but he
was unable to prove that this form is closed without recourse to the
infinite-dimensional picture.  This gap in Goldman's approach was
filled recently by Karshon \cite{ka:algebraic}, who showed that
$d\omega_{\calm }=0 $ by methods of group cohomology,
without using the space of connections.  \footnote{L. Guillou and J.
Huebschmann also inform me
that they have found finite-dimensional proofs that the
form is closed.}

The purpose of this paper is to place Karshon's computations in an
appropriate general setting, namely the double complex of forms on
cartesian powers of $G$ \cite{bo:chern-weil} \cite{sh:characteristic},
or more precisely the corresponding complex of equivariant forms
\cite{at-bo:moment} \cite{be-ge-ve:heat} with respect to the action of
$G$ by conjugation.  


Andreas Floer's refinement of the Casson invariant for homology
3-spheres was constructed in terms of the gradient ``flow'' of the
Chern-Simons invariant on a space of connections.  Floer himself
remarked that there should be an alternative construction of his
invariant which, like Casson's own, takes place on the
finite-dimensional space $\calm $.  \footnote{Some progress in this direction
is reported in \cite{ca-le-mi:symplectic}.}  Such a construction would
use the symplectic structure of $\calm $ in an essential way.  We hope
that the new description of the symplectic structure here may be of
some assistance in the realization of Floer's hope.

I would like to thank S. Bates, R. Bott, A. Givental, Y. Karshon, and
J. Weitsman for their valuable help in the preparation of this paper.  

\section{The de Rham--bar bicomplex}
\label{sec-complex}

The standard ``bar'' complex for computing the cohomology of a group
$G$ \cite{br:cohomology} involves functions on cartesian powers of the
group.  When $G$ is a Lie group, the functions can be replaced by
differential forms, and one obtains a double complex $C^{**}(G)$, with
$C^{p,q}(G)$ defined to be the space $\Omega ^{q}(G^{p})$ of $q$ forms
on $G^{p}$.  The second coboundary operator is just the exterior
differential $d$, while the first, $\delta $, is defined by the usual
formula from group cohomology, with some reinterpretation of the
notation: $$ \label{eq-delta} \delta \alpha (a_{0},\ldots ,a_{p})= $$
$$\alpha (a_{1},\ldots ,a_{p}) + \sum _{i=1}^{p}(-1)^{i}\alpha
(a_{0},\ldots,a_{i-1}a_{i},\ldots,a_{p})+ (-1)^{p+1} \alpha
(a_{0},\ldots ,a_{p-1}) .  $$ Here, when $\phi$ is a differential form
on $Y$ and $y=f(x)$ is the notation ``in variables'' for a mapping
$f:X\rightarrow Y$, the expression $\phi(f(x))$ is interpreted to mean
the pullback of $\phi $ to $X$ by $f$, just as in the case of
functions.

The importance of the de Rham-bar bicomplex revolves around the fact
that the cohomology of the corresponding total complex (with
differential $(-1)^{p}d + \delta )$ is naturally
isomorphic to the real cohomology $H^{*}(BG)$ of the universal
classifying space of $G$.  Furthermore, there is a natural mapping
$\Phi $ from the ring of invariant polynomials on the Lie algebra
$\frakg $ to the total cocycles in $C^{**}(G)$ (see
\cite{sh:characteristic}).

Application of the de Rham--bar bicomplex $C^{**}(G)$ to the moduli
space comes about through the following construction.  Let $\pi $ be a
discrete group.  We consider the space $\hompig$ as a singular
``manifold'' as is usual in this subject \cite{go:symplectic}.  The
evaluation mappings $E: \pi ^{p}\times \hompig \rightarrow G^{p}$
induce pullbacks $E^{*}:C^{p,q}(G)\rightarrow \Omega
^{q}(\pi^{p}\times \hompig )$.  Since $\pi $ is discrete, the last
space can be identified with $C^{p}(\pi )\otimes \Omega ^{q}(\hompig
)$, where $C^{p}(\pi )$ is the usual space of (real-valued)
$p$-cochains on $\pi $, i.e. the functions on $\pi ^{p}$.

Our key observation is that $E^{*}$ is a mapping of bicomplexes, where 
$C^{*}(\pi )\otimes \Omega ^{*}(\hompig )$ is considered as the tensor
product of the bar complex of $\pi $ and the de Rham complex of
$\hompig$.  Composing $E^{*}$ with $\Phi $, we get a map from the
invariant polynomials on $\frakg $ to the tensor product of the
cocycles in $C^{*}(\pi )$ and the closed forms on $\hompig$.  
Applying this map to the quadratic polynomial on $\frakg $
given by an invariant symmetric bilinear form, and pairing the result with the
``fundamental cycle'' for $\pi $, when $\pi $ is the fundamental group
of a Riemann surface, we obtain a closed 2-form $\tilde{\omega }$ on
$\hompig $.   

It turns out that $\tilde{\omega }$ is just the pullback of the
symplectic structure on moduli space.  It is clear from our discussion
so far that $\tilde{\omega }$ is invariant under the adjoint action of
$G$, but it is not so clear that $\tilde{\omega }$ is horizontal.
Horizontality is, however, analogous to closedness where
the operator $d$ is replaced by the operators of interior product with
the infinitesimal generators of the adjoint action.  The sum of these
two operators is essentially the equivariant differential
\cite{at-bo:moment} \cite{be-ge-ve:heat}, so it will be natural to extend
all of our bicomplexes and complexes to their equivariant versions.

\section{Maurer-Cartan calculus}
\label{sec-maurer}

Given a Lie algebra $\frakg $ and a manifold $M$, the $\frakg
$ valued differential forms on $M$ form a super-Lie algebra; i.e. the
algebraic identities defining a Lie algebra are satisfied by $\frakg $
valued forms with the appropriate inclusion of signs \cite{ma:gauge}.  In
particular, $[\eta
,\eta ]$ is not necessarily zero when $\eta $ is a $\frakg$-valued
1-form, but the super Jacobi 
identity implies that $[\eta,[\eta ,\eta ]]=0$.  For further
properties of vector valued differential forms, we refer to
\cite{bi-cr:geometry}.  

On any Lie group $G$, we denote by $\omega$ the $\frakg $-valued left
invariant 1-form which maps each tangent vector to the left
invariant vector field having that value.  The corresponding right
invariant form will be denoted by $\baromega$.    They are related by
the equation
$$ \baromega = \phi\omega , $$
where $\phi $ is the adjoint representation, considered
as a 0-form on $G$ with values in $\mbox{Hom}(\frakg ,\frakg )$.

The forms $\omega $ and $\baromega $ satisfy the structure equations
$$ d\omega =-\half [\omega ,\omega ] $$
and
$$ d\baromega =\half [\baromega ,\baromega ]. $$

Suppose now that $\frakg $ has an Ad-invariant inner product (not
necessarily positive definite), which we
denote simply by $\cdot $~.  Ad-invariance means that $[x,y]\cdot
z=-y\cdot [x,z] $ for any three elements of $\frakg$, so that the
triple product 
\begin{equation}
\label{eq-triple}
[x,y]\cdot z 
\end{equation}
is completely antisymmetric.  For $\frakg
$-valued 1-forms, it is completely symmetric.

We consider the 3-form 
$$ \lambda =\frac{1}{6}\omega \cdot [\omega ,\omega ]. $$
\begin{lemma}
\label{lemma-dlambda}
$d\lambda = 0.$ 
\end{lemma}
\pf 
$[\omega,\omega]$ is closed by the structure equation, so 
$$ d\lambda = -\frac{1}{12}[\omega,\omega ]\cdot [\omega,\omega ].$$
%\pagebreak
By Ad-invariance, the last expression equals $-\frac{1}{12}\omega \cdot
[\omega ,[\omega ,\omega ]]$, which vanishes by the super Jacobi identity.
\qed 

We will use the following notation for forms and vector fields on
$G\times G$.  If $\alpha $ is any differential form on $G$, we denote
by $\alpha _{i}$ the pullback of $\alpha $ to $G\times G$ by the
projection $p_{i}$ onto the $i$'th component.  If $X$ is a
vector field on $G$, we denote by $X_{i}$ the vector field on $G\times
G$ which projects to $X$ under $p_{i}$ and to zero under the other
projection.  $m:G\times G\rightarrow G$ is group multiplication.


In terms of this notation, we have the following lemma, whose proof is
left to the reader:

\begin{lemma}
\label{lemma-maurer}
$m^{*}\omega = \phi _{2}^{-1} \omega _{1}+\omega _{2}.$
\end{lemma} 

The object which will eventually give us the symplectic structure on
moduli space is the 2-form
$$ \Omega =\omega _{1}\cdot \baromega _{2} $$
on $G\times G$.  
Using the structure equations and the relation between $\omega $ and
$\baromega $, we compute the exterior derivative
$$d\Omega = -\half([\omega _{1},\omega _{1}]\cdot \phi_{2} \omega _{2} 
+ \omega _{1}\cdot [\phi_{2} \omega _{2},\phi_{2} \omega _{2}]). $$ 

The form $d\Omega $ is evidently nonzero (unless $G$ is abelian).
However, we have: 
\begin{lemma}
\label{lemma-domega}
$$ d\Omega = \delta \lambda .$$
\end{lemma}
\pf 
$$ -\delta \lambda = m^{*} \lambda -\lambda _{1} -\lambda _{2} = $$
$$ \frac{1}{6}(\phi _{2}^{-1}\omega _{1}+\omega _{2})\cdot [\phi _{2}^{-1}\omega _{1}+\omega
_{2},\phi _{2}^{-1}\omega _{1}+\omega _{2}] - \frac{1}{6}\omega_{1} \cdot [\omega_{1}
,\omega_{1}]-\frac{1}{6}\omega_{2} \cdot [\omega_{2} ,\omega_{2} ]. $$ 
Using the fact that the values of $\phi $ are automorphisms of both
the bracket and inner product structures on $\frakg $, we may simplify
the expression above to:
$$ \half ([\omega _{1},\omega _{1}]\cdot \phi_{2}\omega_{2}  
   + \omega _{1}\cdot
[\phi_{2}\omega_{2},\phi_{2}\omega_{2}]). $$ 
\qed 

Another simple computation, which we leave to the reader, shows:

\begin{lemma}
\label{lemma-deltaomega}
$\delta \Omega =0.$
\end{lemma}

Combining Lemmas \ref{lemma-dlambda}, \ref{lemma-domega}, and
\ref{lemma-deltaomega}, we conclude:

\begin{cor}
\label{cor-omegaminuslambda}
$\Omega - \lambda $ is a cocycle of degree 4 in the total complex of
the double complex $C^{**}(G)$.
\end{cor}

%\begin{rmk}
%\label{rmk-bott-shulman}
\noindent
{\bf  Remark}~
The forms $\Omega $ and $\lambda $ were originally found by analysis
of objects in \cite{ka:algebraic}, but in fact $\Omega -\lambda $ is
precisely the result of applying the Bott-Shulman map $\Phi $ to the
invariant polynomial $X\cdot X$ on $\frakg $.
%\end{rmk}



\section{The equivariant theory}
\label{sec-equivariant}

The equivariant cohomology of a $G$-manifold $M$ can be computed using the
complex of {\em equivariant differential forms} on $M$
\cite{at-bo:moment} \cite{be-ge-ve:heat}.  By
definition, these are polynomial maps from $\frakg $ to $\Omega
^{*}(M)$ which are equivariant with respect to the adjoint
representation.  The equivariant differential $d_{G}$ is defined by
$(d_{G}\alpha)(X)=(d-i_{X_{M}})(\alpha(X))$.  Here, $X_{M}$ denotes
the vector field given by the Lie algebra element $X$
acting on $M$.  

In the setting of the previous section, if a group $H$ acts on $G$ by
automorphisms, it is natural to extend the double complex $C
^{**}(G)$ to its equivariant version $C^{**}_{H}(G)$, where
$C^{p,q}_{H}(G)$ consists now of the $H$-equivariant $q$-forms
on $G^{p}$, and the second differential is replaced by $d_{H}$.
\footnote{We expect that the cohomology of the corresponding total
complex should be isomorphic to the $H$-equivariant cohomology of
$BG$, which is in turn isomorphic to the cohomology of $BK$, where $K$
is the semidirect product of $H$ with $G$.}  In what follows,
we will restrict our attention to the special case where $H=G$, acting
by inner automorphisms. 

The form $\Omega - \lambda $, which is closed for the operator $\delta
+(-1)^{p}d$, is {\em not} equivariantly closed.  The extra part of the
equivariant differential is calculated in the next two lemmas, in
which the $X$ on the right hand side is to be interpreted as a
constant $\frakg $-valued 0-form on G.

\begin{lemma}
\label{lemma-ixlambda}
$i_{X_{G}}\lambda = -d(X\cdot (\omega +\baromega ))$. 
\end{lemma}
\pf We begin by noting that $X_{G}$ for the adjoint representation is
just $X-\overline{X}$, where $\overline{X}$ is the right-invariant
vector field which agrees at the identity with the left invariant
vector field  $X$.  Then we have
$$ 6i_{X_{G}}\lambda=
i_{X-\overline{X}}(\omega \cdot [\omega,\omega ]). $$   
Since $\lambda $ is also equal to $\baromega \cdot
[\baromega,\baromega ]$, the last expression can be rewritten as 
$$ i_{X}(\omega \cdot [\omega,\omega ])
-i_{\overline{X}}(\baromega \cdot [\baromega,\baromega ]). $$
Using the equations $i_{X}\omega =X$ and $i_{\overline{X}}\baromega
=X$ and the symmetry properties of the triple product (\ref{eq-triple}),
the last expression simplifies to 
$$ 3X\cdot ([\omega ,\omega ]-[\baromega ,\baromega ]), $$
which, by the structure equations, is just 
$$ -d(6X\cdot (\omega +\baromega )).$$
\qed 

\begin{lemma}
\label{lemma-ixomega}
$i_{X_{G\times G}}\Omega = X\cdot (\omega _{1}- \phi _{2}^{-1}
\omega _{1} +\baromega _{2}-\phi_{1}\baromega _{2})$. 
\end{lemma}
\pf 
Since $X_{G\times G}=X_{1}-\overline{X}_{1}+X_{2}-\overline{X}_{2}$,
we have
$$ i_{X_{G\times G}}\Omega
=i_{X_{1}-\overline{X}_{1}+X_{2}-\overline{X}_{2}}(\omega _{1}\cdot
\baromega _{2}) $$
Using the fact that $\overline{X}=\phi ^{-1}X$, one can easily
transform the last expression into the
right hand side of the statement of the lemma.
\qed 
 
Lemmas \ref{lemma-ixlambda} and \ref{lemma-ixomega} suggest that a
piece of the puzzle is missing, namely the equivariant form 
$$ \theta (X)=X\cdot (\omega +\baromega). $$  The calculations above
show that $$(\delta +(-1)^{p}d_{G})(\Omega -\lambda +\theta )(X)=-i_{X_{G\times
G}}\Omega +\delta \theta(X) + i_{X_{G}}\theta (X).$$ The next two lemmas now
complete the picture.  
\begin{lemma}
\label{lemma-ixtheta}
$i_{X_{G}}\theta (X)=0$.
\end{lemma}
\pf 
$$ i_{X_{G}}\theta (X)=i_{X-\overline{X}}(X\cdot (\omega +\baromega) )=
X\cdot (i_{X}\omega -i_{\overline{X}}\baromega )=0. $$
\qed 
\begin{lemma}
\label{lemma-deltatheta}
$\delta \theta(X) =i_{X_{G\times G}}\Omega $
\end{lemma}
\pf 
$$\delta \theta(X) =  \theta(X)_{1} + \theta(X)_{2}
-m^{*}\theta(X)$$
$$ =X\cdot (\omega _{1}+\baromega _{1}+\omega _{2}+\baromega _{2}-
  (\phi _{2}^{-1} \omega _{1}+\omega _{2}+\baromega _{1}+\phi
_{1}\baromega _{2}) ) $$
$$ =X\cdot (\omega _{1}+\baromega _{2}-  (\phi _{2}^{-1} \omega _{1}+\phi
_{1}\baromega _{2}) ). $$
Now compare with Lemma \ref{lemma-ixomega}.
\qed 
As a consequence of the preceding calculations, we have

$$ (\delta +(-1)^{p}d_{G})(\Omega -\lambda +\theta )= 0.$$  We state
our main results below.

\begin{thm}
\label{thm-main}
$-\lambda +\theta $ is an equivariantly closed form of degree 3 on
$G$, and $Q_{4}=\Omega -\lambda +\theta $ is an equivariantly closed
element of degree 4 in the total complex of the bicomplex
$C_{G}^{**}(G)$.  
\end{thm}

\section{Forms on representation spaces}
\label{sec-representation}
To complete our work, we have little to do but to pull back the 4-form
$Q_{4}$ of Theorem \ref{thm-main} by the evaluation maps $E: \pi
^{p}\times \hompig \rightarrow G^{p}$.  Since each $E$ is obviously
equivariant, $E^{*}Q_{4}$ is a cocycle in the total complex of the
bicomplex $C^{*}(\pi )\otimes C_{G}^{*}(\hompig )$.  By pairing
with cycles in $C_{*}(\pi )$, we obtain equivariant forms on
$\hompig $.  

If $c\in C_{2}(\pi)$ is a 2-cycle, then 
$$ d_{G}\langle E^{*}\Omega ,c\rangle =d_{G}\langle E^{*}Q_{4},c\rangle =\langle E^{*}d_{G}Q_{4},c\rangle 
$$  $$=E^{*}\langle -\delta
Q_{4},c\rangle =-E^{*}\langle Q_{4},\del c\rangle =0, $$  
so $\langle E^*\Omega,c \rangle$ is an equivariantly closed 2-form on
$\hompig $.   A
similar computation using $\Omega -\lambda $ and the ordinary
differential shows that $\langle E^*\Omega,c \rangle $ is closed.  It follows that
$\langle E^*\Omega,c \rangle$ is a $G$-invariant closed form on $\hompig$ which is
annihilated by $i_{X_{\rm{Hom}(\pi ,G)}}$ for every $X\in \frakg $, so that it
is the pullback of a closed 2-form $\omega _{\calm }$ on (at least the
smooth part of) the moduli space $\calm $.  

If we pair with a 2-boundary $\del b$, then we have $$ \langle
E^{*}\Omega,\del b\rangle =\langle \delta E^{*}\Omega ,b\rangle
=\langle E^{*}\delta \Omega ,b\rangle ,$$ which is zero by Lemma
\ref{lemma-deltaomega}.  Thus we have proven:

\begin{thm}
\label{thm-twoform}
Pairing $E^{*}\Omega $ with 2-cycles defines a natural
homomorphism from $H_{2}(\pi )$ to the space of closed 2-forms on
$\calm $.
\end{thm}

Following \cite{ka:algebraic}, one may show easily that if $\pi $ is
the fundamental group of a closed Riemann surface and $c$ is the
fundamental cycle of that surface, then $\omega _{\calm }$ is the
usual symplectic structure on moduli space.

It is also possible to pair $E^{*}(\lambda )$ with a class of degree
$1$ in $H_{*}(\pi )$ (i.e. a conjugacy class in $\pi $).  The
resulting object is a closed, invariant, 3-form on $\hompig $.  This
form does not, however, push down to $\calm $.  In fact, if we
subtract $E^{*}\theta $ from $E^{*}\lambda $, the result is an
equivariantly closed form on $\hompig $, but it can be pushed down as
a class to
$\calm $ only where the $G$ action is free, and as a form only with the aid
of a connection for that action.

A reader familiar with other work on the cohomology of moduli spaces
may recognize that we have found, in a new guise, the generating
classes of Newstead \cite{ne:characteristic}.  (Actually, these
classes only apply directly to the ``twisted'' moduli spaces; see
\cite{at-bo:yang-mills}.)  One of Newstead's generators is, however,
missing, namely the class of degree 4 obtained by pairing with the
fundamental 0-cycle of $\pi $.  In fact, we can obtain this class as
well by considering the element of $C_{G}^{0,4}(G)$ given simply by
the invariant quadratic polynomial $\Phi (X)=X\cdot X$.  This pulls
back to $\Omega _{G}^{*}(\hompig )$ as the same polynomial, which is
an equivariant form of degree 4.  The limitations on pushing down to
$\calm $ are the same as those just discussed for the classes of
degree 3.

It is perhaps useful to describe the symplectic form on $\calm$, as we
have constructed it, in more concrete terms.  A 2-cycle for $\pi$ is
just a (finite) formal linear combination of elements of
$\pi\times\pi$.  Pairing $E^*(\Omega)$ with such a cycle gives a
linear combination of pullbacks of $\Omega$ from $G^2$ to $\hompig$ by
evaluation on pairs of elements of $\pi$.  Now choose a finite subset
$\gamma$ of $\pi$ which contains a generating set and is large enough so
that our cycle is supported in $\gamma\times\gamma$.  We consider the
singular space $\hompig$ as embedded into the smooth manifold
$G^{\gamma}$ of maps from $\gamma$ to $G$.  The evaluation map $E$
extends in an obvious way to a map from $\pi^2 \times G^{\gamma}$ to
$G^{2}$, so when we pair with the cycle we get a smooth 2-form on
$G^{\gamma}$ which extends our form on $\hompig$.  It is important to
note, though, that the extended evaluation map is no longer compatible
with the two bicomplex structures, so {\em the extended 2-form is
not necessarily closed}.  We do not know whether a smooth closed
extension can be found; nevertheless, it is reassuring to see the form
on the singular space $\hompig$ extended to a smooth form on a smooth
manifold.

\section{Discussion}
\label{sec-discussion}
Our construction of the symplectic structures on moduli spaces raises
several questions.

First of all, it would be nice to prove with our
formalism that the closed 2-form on moduli space is nondegenerate.
One way to do so would be to construct the corresponding Poisson
structure by an analogous procedure.  This approach leads to the
idea of applying our method to the moduli spaces of flat
connections on manifolds with boundary, which are Poisson manifolds in
which the symplectic leaves are given by specifying the conjugacy
class of the holonomy on each boundary circle.

Secondly, we have dealt with only the characteristic class of degree 4
for the group $G$.  It turns out that this leads to several
equivariant cohomology classes on $\hompig $, only one of which passes
to a canonical form on $\calm $.  It would be interesting to carry out
a similar analysis for the entire characteristic ring of $G$.  

Although the forms $\Omega $ and $\lambda $ were seen to
arise directly by the Bott-Shulman construction, the equivariant form
$\theta (X)$ was put in ``by hand''.  It should be possible to find a
``grand unified theory'' encompassing all three of these forms, as
well as the degree 4 class mentioned near the end of Section
\ref{sec-representation}, by using a suitable de Rham model for
$H^{*}_{G}(BG).$ 

The geometric quantization of $\calm $, which is so important for
topological quantum field theory, begins with the construction of a
line bundle whose curvature is $\omega _{\calm }$.  The construction
of this line bundle in \cite{ra-si-we:comments} relies on the
Chern-Simons functional.  Is it possible to construct the line bundle
from an object analogous to the de Rham--bar complex?

We hope to return to these points in a sequel to this paper.

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