Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

%Intersection pairings in moduli spaces of 
%holomorphic bundles on a Riemann surface
%by Lisa Jeffrey and Frances Kirwan


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\begin{document}

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\title{Intersection pairings in moduli spaces of holomorphic bundles 
on a Riemann surface}
\author{Lisa C. Jeffrey 
\and 
Frances C. Kirwan}
\address{Lisa C. Jeffrey, Mathematics Department, Princeton University, Princeton,\newline NJ 08544, USA}
\email{jeffrey@@math.princeton.edu}
\address{Frances C. Kirwan, Balliol College, Oxford OX1 3BJ, UK}
\email{fkirwan@@vax.ox.ac.uk}
\thanks{This material is based
on work supported by the National Science Foundation under
Grant. No. DMS-9306029.}
\subjclass{58F05, 14F05, 53C05}
\keywords{Moduli spaces, symplectic geometry, intersection pairings}
\date{June 28, 1995}
\begin{abstract}
 We outline a proof of formulas (found by Witten in 1992
	using physical methods) for intersection pairings in the 
	cohomology of the moduli space $M(n,d)$ of stable holomorphic 
	vector bundles of rank $n$ and degree $d$ (assumed coprime) and 
	fixed determinant on a Riemann surface of genus $g \ge 2$.
\end{abstract}
\maketitle


\section{Introduction}
\markboth{LISA C. JEFFREY AND FRANCES C. KIRWAN}{INTERSECTION PAIRINGS IN MODULI SPACES}

The moduli space $M(n,d)$ of semistable rank $n$ degree $d$ holomorphic
vector bundles with fixed determinant on a compact Riemann surface $\Sigma$
is a smooth 
K\"ahler manifold when $n$ and $d$ are coprime. This space had long
been studied by algebraic geometers (see for instance
Narasimhan and Seshadri 1965 \cite{NS}), but a new viewpoint on it was
revealed 
by the seminal 1982
 paper \cite{AB} of Atiyah and Bott on the Yang-Mills equations
on Riemann surfaces. In this paper a set of generators for the  (rational)
cohomology ring of $M(n,d)$ was given and formulas for
the Poincar\'e polynomial were proved. 
Given the specification of a set of generators,
knowledge of the intersection pairings between products of these
generators (or equivalently knowledge
of the evaluation on the fundamental class of products of 
the generators) 
completely determines the structure of the cohomology ring.

Little progress was made on the problem of determining these intersection 
pairings until 1991, when Donaldson \cite{Do} and Thaddeus \cite{T} 
gave formulas for the 
intersection pairings in $H^*(M(2,1)) $ (in terms of Bernoulli
numbers). Then, using physical methods, Witten \cite{tdgr}
found formulas for the evaluation of any product  of  generators
on the fundamental class of $M(n,d)$ for arbitrary rank $n$. 
These generalized his (rigorously 
proved) formulas \cite{qym} for the symplectic volume of $M(n,d)$:
 for instance, the symplectic 
volume of $M(2,1)$ is given by 
 $$ {\rm vol}(M(2,1)) = 2 \frac{(-1)^{g-1} }{(2 \pi^2)^{g-1} }
 \Bigl  (1 - \frac{1}{2^{2g-3}} \Bigr )  \zeta(2g-2)$$ where
$g$ is the genus of the Riemann surface and $\zeta$ is the Riemann 
zeta function.

In this article we 
outline a mathematically rigorous proof of Witten's result. Full
details, in particular for the case of rank $n$ at least three,
will appear later \cite{JK2}. The key
idea is to use a description of $M(n,d)$ as a symplectic
quotient and to apply the {\it nonabelian localization theorem}
(Witten \cite{tdgr}; Jeffrey-Kirwan \cite{JK1}), 
which is a generalization of the
Duistermaat-Heckman theorem \cite{DH}. A different approach to the nonabelian
localization principle has been given recently by Guillemin-Kalkman 
\cite{GK}
and independently by Martin \cite{Ma}; their ideas are crucial in the
technicalities of our proof since they can be adapted to certain noncompact
situations where the approach of \cite{JK1} is not valid.


This paper is organized as follows. In Section 2 we describe the generators
for the cohomology ring $H^*(\mnd)$. In Section 3 we outline tools from 
the Cartan model of 
equivariant cohomology and
some different versions of localization.
In Section 4 we recall properties of 
 the {\em extended moduli space}, a finite dimensional 
symplectic space equipped with a Hamiltonian action of $SU(n)$ for which
the symplectic quotient is $\mnd$. Finally Section 5 gives an outline
of the proof of 
Witten's formulas.






\section{Generators for the cohomology ring}
Let us assume throughout, in order to avoid exceptional
cases, that the Riemann surface $\Sigma$ has genus $g\geq 2$.

In \cite{AB}, Atiyah and Bott gave a set of generators
for $H^*(\mnd)$; their procedure may be described as follows\footnote{In this 
paper, all cohomology groups are assumed to be with complex
coefficients.}.
 We may form a normalized universal rank $n$ vector  bundle
$$ \UU \to \Sigma \times \mnd $$ (see \cite{AB}, p. 582). 
 Then there is the following description \cite{AB} of a set of 
generators of $H^*(\mnd)$: 
\begin{equation} f_r = ([\Sigma], c_r(\UU)), \end{equation}
\begin{equation} b_r^j = (\a_j, c_r(\UU)), \end{equation}
\begin{equation} a_r = (1, c_r(\UU)). \end{equation}
Here, $[\Sigma]$
 $ \in H_2(\Sigma)$ and $\a_j \in H_1(\Sigma)$ $(j = 1, \dots, 2g)$
form standard bases  of $H_2(\Sigma, \ZZ)$ and  $H_1(\Sigma, \ZZ)$. 
The bracket represents the slant product $H^N(\Sigma \times \mnd)
\otimes H_j(\Sigma) \to H^{N-j}(\mnd). $ 
 More generally if $K=SU(n)$ and $Q$ is an invariant polynomial of
degree $s$ on its Lie algebra $\liek = su(n)$ then there is an
associated element of $H^*(BSU(n))$ and hence an associated
element of $H^*(\Sigma \times \mnd)$ which is a characteristic
class $Q(\UU)$ of the universal bundle $\UU$. Hence the slant
product gives rise to classes 
\begin{equation} Q([\Sigma]) \in H^{2s-2}(\mnd), \end{equation}
\begin{equation} Q(\alpha_j) \in H^{2s-1}(\mnd), \end{equation}
\begin{equation} Q(1) \in H^{2s}(\mnd). \end{equation}
In particular, letting $\tau_r$ $\in S^r(\lieks)^K$ 
denote the invariant polynomial associated to the 
$r$-th Chern class, we recover our generators 
of $H^*(\mnd)$: 
\begin{equation} \label{1.2}
f_r = \tau_r([\Sigma]), \end{equation}  
$$ b_r^j = \tau_r (\a_j), $$
$$ a_r = \tau_r(1). $$
A special role is played by the invariant polynomial 
$\tau_2 = \frac{1}{2}\inpr{\cdot , \cdot} $  on $\liek$
given by an invariant 
inner product proportional to 
the Killing form.  We normalize the inner product
as follows for $K = SU(n)$: 
$$ \inpr{X, X} = - {\rm Trace} (X^2)/(4 \pi^2). $$
The associated class $f_2$ is the cohomology class
of the symplectic form on $\mnd$.

In 
 Sections 4 and 5
of \cite{tdgr}, Witten obtained formulas for generating functionals 
from which one may extract all intersection pairings
$$ \prod_{r = 2}^n a_r^{m_r} f_r^{n_r} \prod_{k_r = 1}^{2g} 
(b_r^{k_r})^{p_{r, k_r} } [\mnd].$$

Let us begin with pairings of the form
\begin{equation} \label{1.00001}
\prod_{r = 2}^n a_r^{m_r} \exp f_2 [\mnd]. \end{equation}
For $m_r$ sufficiently small  to ensure convergence
of the sum,  Witten obtains
(\cite{tdgr}, (4.74))
\begin{equation} \label{1.1}
\prod_{r = 2}^n a_r^{m_r} \exp f_2 [\mnd]
= 
\G   \Biggl ( 
\sum_{\l  } \frac{c^{- \l} 
\prod_{r = 2}^n \tau_r(2 \pi i\l) ^{m_r} }
{\nusym^{2g-2}(2 \pi i \l) } \Biggr ),  \end{equation}
where
$\G
$ 
 is a universal constant depending on $n$, $d$  and $g$, and
the Weyl odd polynomial 
$\nusym$ on $\liets$ is defined by 
$$\nusym(X) = \prod_{\g > 0 } \g(X) $$
where $\g$ runs over the positive roots.
The sum over $\l$ in 
(\ref{1.1}) 
runs over those elements of the weight 
lattice $\weightl$
that are in the interior of the fundamental Weyl chamber.\footnote{The weight
lattice $\weightl \subset \liet$ is the dual lattice of the integer
lattice $\intlat = {\rm Ker} (\exp) $ in $\liet$ under 
the inner product $\langle \cdot, \cdot \rangle$.}
The 
element 
\begin{equation} \label{1.p1} c = e^{2 \pi \isq  d/n}
{\rm diag}( 1, \dots, 1) \end{equation}
is a generator of the 
centre $Z(K)$, so since $\l \in  \liets$ is in 
${\rm Hom} (T, U(1))$, we may evaluate $\l$ on $c$ as in (\ref{1.1}): 
$c^\l$ is defined as $\exp \l (\tilde{c}) $ where $\tilde{c}$ is 
any element of the Lie algebra of $T$ such that 
$\exp \tilde{c} = c$. 




There are similar formulas 
(\cite{tdgr}, (5.15) and (5.18))
for pairings involving the 
$f_r $ for $r > 2 $ and  the $b_r^j$ as well as $f_2$ and
the $a_r$. 



For concreteness it is worth examining the special case $n = 2, d = 1$. 
Here the dominant weights $\l$ are just the positive integers.
The relevant generators of $H^*(\mto)$ are \begin{equation} f_2 \in H^2(\mto)\end{equation}
(which is the cohomology class of the symplectic form 
on $\mto$)
and 
\begin{equation} a_2 \in H^4(\mto): \end{equation}
 these arise from the invariant polynomial 
$\tau_2 = \inpr{\cdot,\cdot}$ by 
$a_2 = \tau_2(1)$, $f_2 = \tau_2([\Sigma]) $ 
(see (\ref{1.2})). We find then that the formula (\ref{1.1}) reduces for 
$m \le g - 2 $ to\footnote{Here, we have identified $a_2$ with Witten's class
$\Theta$ and $f_2 $ with Witten's class $\omega$.}
(\cite{tdgr}, (4.44))
\begin{equation} \label{1.3}
a_2^j \exp (f_2 )[{\calm}(2,1)] 
= \frac{2^{2g}(-1)^{g-1-j} }{2 (8 \pi^2)^{g-1}}
 \Biggl ( \sum_{n = 1}^\infty 
(-1)^{n + 1}\frac{  \pi^{2j}  }{ n^{2g-2-2j }}
\Biggr ). \end{equation} 
 Thus one obtains the 
  formulas   found in \cite{T} for the intersection pairings \newline
$a_2^m f_2^n [\mto]$; these intersection pairings are given by 
Bernoulli numbers, or equivalently are given in terms of the 
Riemann zeta function $\zeta(s) = \sum_{n \ge 1} 1/n^s$. 


\section{Equivariant cohomology and localization}

The methods we shall use involve the application of the 
{\em nonabelian localization theorem} \cite{JK1,tdgr}.
In \cite{JK1} we considered a compact symplectic manifold
$M$ equipped with the Hamiltonian action of a compact 
group $K$. We expressed the cohomology ring 
$H^*(\xred)$ of the reduced space or symplectic 
quotient $\xred$ $= \mu^{-1}(0)/K$
 of $M$ by $K$ in terms of  the equivariant cohomology
 of $M$. (This was under the assumption that $0$ is a regular
value of $\mu$, which implies that $\xred$ is a symplectic orbifold.) 
There is a surjective ring homomorphism
$\Phi$ from the equivariant cohomology $\hk(M)$ of $M$ to 
the cohomology $H^*(\xred)$ of the reduced space. In terms of this map, 
we derived a formula for the evaluation 
$\eta_0 [\xred]$ of a cohomology class $\eta_0 \in H^*(\xred)$ 
on the fundamental class of $\xred$. This formula 
involves data that   enter the Duistermaat-Heckman formula,
and its generalization the abelian localization formula
\cite{ABMM,BV1,BV2}
 for the action of a maximal torus
$T$ of $K$ on $M$: that is, the components $F$ of the fixed point set
$M^T$ of $T$ on $M$, and the equivariant Euler classes $e_F$ of the normal 
bundles to $F$ in $M$. In the following paragraphs we sketch this
construction.

The $K$-equivariant cohomology of $M$ is the cohomology of a certain 
chain complex $\Omega^*_K(M)$, which can be expressed as 
\begin{equation} \label{1.001} \Omega^*_K(M) = (S(\lieks) \otimes \Omega^*(M))^K \end{equation}
(where $\Omega^*(M)$ denotes the space of 
differential forms on $M$, and $S(\lieks)$ denotes the ring
of polynomial functions on the Lie algebra $\liek$ of $K$). An element 
$f \in \Omega^*_K(M)$ may be thought of 
as a $K$-equivariant polynomial function from $\liek$ 
to $\Omega^*(M)$, or alternatively as a family of differential forms
on $M$ parametrized by $\xvec \in \liek$. 
The differential $D$ on the complex $\Omega^*_K(M)$ is then defined by 
\begin{equation} \label{1.0002}
(D\alpha)(\xvec) = d(\alpha (\xvec) )  - \iota_{\xvec^{\#}} (\alpha(\xvec) ) \end{equation}
where $\xvec^{\#}$ is the vector field on $M$ generated by the action of 
$\xvec$. 

In terms of this notation, the map $\Omega^*_K(M) \to 
\Omega^*_K({\rm pt}) = S(\lieks)^K$ 
given by integration over $M$ passes to $\hk(M)$. Thus  for any 
$D$-closed element 
$\eta \in \Omega^*_K(M)$ representing a cohomology class $[\eta]$, 
there is a corresponding element 
$\int_M \eta \in \Omega^*_K({\rm pt})$ which depends only on 
$[\eta]$. The same is true for any 
$D$-closed element $\eta = \sum_j \eta_j$ 
 which is a formal 
 series of elements $\eta_j$ in $\Omega^j_K(M)$  without polynomial 
dependence on $\xvec$: we shall in particular consider terms
of the form 
$\eta (\xvec) e^{\isq (\omega + \mu(\xvec))} $ (where 
$\eta \in \Omega^*_K(M)$), since the element 
$$\bom(\xvec) = \omega + \mu(\xvec) \in \Omega^2_K(M) $$
satisfies $D \bom = 0 $. 

 If $\xvec$ lies in $\liet$, the Lie algebra of 
a chosen maximal torus $T$ of $K$, then there is a formula for 
$\int_M \eta(\xvec)$ (the {\em   abelian localization 
formula \cite{AB,BGV,BV1,BV2}})
which depends only on the fixed point set of 
$T$ in $M$. It tells us that 
\begin{equation} \label{1.002}
 \int_M \eta(\xvec) 
= \sum_{F \in \calf} \int \frac{i_F^* \eta(\xvec)}
{e_F(\xvec)} \end{equation}
where $\calf$ indexes the components $F$ of the fixed point set of 
$T$ in $M$, the inclusion of $F$ in $M$ is denoted
by $i_F$ and $e_F $ 
$\in H^*_T(M)$ is the equivariant Euler class of the normal 
bundle to $F$ in $M$. In particular, applying (\ref{1.002})
with  $\eta $ replaced by the formal sum $\eta e^{\isq \bom}$ 
we have 
\begin{equation} \label{1.003} h^\eta(\xvec) \eqdef
 \int_M \eta(\xvec)e^{\isq \bom(\xvec)}  
= \sum_{F \in \calf} h^\eta_F(\xvec), \end{equation}
where 
\begin{equation} \label{1.004}   \hfeta(\xvec) = 
e^{\isq \mu(F)(\xvec)}\int_F  \frac{i_F^* \eta(\xvec)  e^{\isq \omega} 
 }{e_F (\xvec) }. \end{equation}
Note that the moment map $\mu$ takes a 
constant value $\mu(F) \in\liets$ 
 on each 
$F \in \calf$, and that the integral in (\ref{1.004}) 
is a rational function of $\xvec$. 

\nc{\res}{{\rm Res}}
We assume that  $\eta_0 \in H^*(\xred)$ comes  
via the surjective homomorphism $\Phi$ from a class in 
$\hk(M) $ which is represented as an equivariant 
differential form by  $\eta \in \Omega^*_K(M)$, 
and denote the symplectic form on 
$\xred$ by $\omega_0$. 
Let us recall   the main result (the residue formula, Theorem 
8.1) of \cite{JK1}: 
\begin{theorem} \label{t4.1}{\cite{JK1}} Let $\eta \in 
\hk(M) $ induce $\eta_0 \in H^*(\xred)$. Then we have 
\begin{equation}   \eta_0 e^{{i}\omega_0} [\xred] 
 = {n_0 C^K}  \res \Biggl ( 
\nusym^2 (X)
 \sum_{F \in \calf} \hfeta(X) [d X] \Biggr ), \end{equation}
where $n_0$ is the
order of the subgroup of $K$ that acts trivially on\footnote{The subgroup of
$K$ which acts trivially on $M$ is the same as the subgroup, $N$ say, of $K$ which
acts trivially on $\zloc$, and is a finite central subgroup of $K$. For since $0$ is
a regular value of $\mu$ it follows that $N$ is a finite normal subgroup of
$K$, and because $K$ is connected $N$ is therefore contained in the center
of $K$. Thus the coadjoint action of $N$ on $\lieks$ is trivial, so $N$ acts trivially on the
normal bundle to $\zloc$ in $M$, and as $M$ is connected this means
$N$ acts trivially on $M$.}
$M$, and  
the constant $C^K$ is defined by 
\begin{equation} \label{4.001} C^K = \frac{(-1)^{n_+}}{  |W| \vol(T)}. \end{equation}
 We have introduced $s = \dim K$ and 
$l  = \dim T$; here
$n_+ = (s-l)/2$ is the 
number of positive roots\footnote{Here, the roots of $K$ are the 
nonzero weights of its complexified adjoint action. We fix
the convention that weights $\beta \in \liet^*$ satisfy
$\beta \in {\rm Hom}
(\Lambda^I,\ZZ)$ rather than 
$\beta \in {\rm Hom} (\Lambda^I, 2 \pi \ZZ)$ (where $\Lambda^I 
= {\rm Ker}(\exp : \liet \to T)$ is the integer lattice). This 
definition of roots differs by a factor of $2 \pi$ from the definition
used in \cite{JK1}: there, the roots $\gamma$ satisfy
$\gamma(\Lambda^I) \subset 2 \pi \ZZ$. This is why the constant
$C^K$ differs from that of \cite{JK1} by a factor of $(2\pi)^{s-l}$.}. 
Also, $\calf$ denotes the set of components of the fixed point
set of $T$, and if $F$ is one 
of these components then the meromorphic  function 
$\hfeta$ on $\liet \otimes \CC$ is defined by 
(\ref{1.004}) and 
the polynomial
$\nusym: \liet \to \RR$ is defined by 
$\nusym(X) = \prod_{\g > 0 } \g(X) $, where $\g$ runs 
over the positive roots of $K$. 
\end{theorem}
The above formula was called a residue formula in \cite{JK1} because the 
quantity $\res$  
(whose general definition 
was given in Section 8 of \cite{JK1})
can be expressed as a multivariable
residue (or alternatively in terms of iterated 1-variable residues). 

 Here we shall make particular
use of 
the case where $K$ has rank $1$, for which the results are as 
follows.  %See  Footnotes 2 and 3   for our conventions
%on roots and weights.
\begin{corollary} \label{c4.2} 
In the 
situation of Theorem \ref{t4.1}, let $K = U(1)$.
 Then
$$ \eta_0 e^{{i}\omega_0} [\xred] = 
{n_0}  \res_{X=0} \Bigl (  \sum_{F \in \calf_+} \hfeta(X) 
\Bigr ) . $$ 
Here, the meromorphic  function $\hfeta $ on  $ \CC$
 was defined 
by (\ref{1.004}), and $\res_{X=0} $ denotes the coefficient
of $1/X$,  where 
$X \in \RR$ has been identified with $2 \pi i X \in \liek$.  
    The set $\calf_+ $ is defined by 
$\calf_+ = \{ F \in \calf: \mu_T(F) > 0 \}. $ The integer
$n_0$ is as in Theorem \ref{t4.1}.
\end{corollary}

\begin{corollary} \label{c4.3}{\bf(cf. \cite{JK1}, Corollary 8.2)}
In the 
situation of Theorem \ref{t4.1},
let $K = SU(2)$. Then 
$$ \eta_0 e^{{i}\omega_0} [\xred] = 
 -\frac{n_0 }{2} \res_{X=0} \Bigl (  (2X)^2  \sum_{F \in \calf_+} \hfeta(X)
  \Bigr )  . $$ 
Here, $\res_{X=0}$, $\hfeta$ and $\calf_+$ are as in Corollary 
\ref{c4.2}, and $X \in \RR $ has been identified with
${\rm diag} (2 \pi i, - 2 \pi i) X \in \liet$. 
The integer
$n_0$ is as in Theorem \ref{t4.1}. 
\end{corollary}

  



\noindent{\em Remark:} Notice that
 we shall not only be considering the 
reduced space $\xred = \mu^{-1}(0)/K$ with respect to the action of 
the nonabelian group $K$, but  also 
 $\mu^{-1}(0)/T$ and 
\nc{\mredt}[1]{M_{\rm red}^T(#1)} 
$\mredt{t}$  $ = \mu_T^{-1}(t)/T$ 
for regular values $t$ of the $T$-moment map $\mu_T$ which is the 
composition of $\mu$ with restriction from $\lieks$ to $\liets$. 
We shall use the same notation $\eta_0$ for the image of $\eta$
under the surjective homomorphism
$\Phi$ for whichever of the spaces
$\mu^{-1}(0)/K, $ $\mu^{-1}(0)/T $ or
$\mu_T^{-1}(0)/T $ we are working with, and the 
notation $\eta_t$ if we are working with $\mu_T^{-1}(t)/T$. 
It should be clear from the context which version of the map 
$\Phi$ is being used.







We shall need the following very recent   additional results, which 
are expressed in the above notation:

\begin{prop}  \label{p:sm}
{(Reduction to the abelian case)} ({\sc S. Martin 
\cite{Ma}}) 
If $T$ is a maximal torus of $K$, then we have that 
$$ \int_{\mu_K^{-1}(0)/K} 
(\eta e^{i \bom} )_0  = \frac{1}{|W|}
\int_{\mu_K^{-1}(0)/T} 
(\nusym \eta e^{i \bom} )_0 = \frac{(-1)^{n_+} }{|W|}
\int_{\mu_T^{-1}(t)/T}
(\nusym^2 \eta e^{i \bom} )_t $$
for any regular value $t$ of $\mu_T$ sufficiently
close to $0$.  
\end{prop}

\nc{\mtred}{M^T_{\rm red} }

\begin{prop}  \label{p:gkm}
{(Dependence  of symplectic quotients on parameters)} 
({\sc Guille\-min-Kalkman \cite{GK} ; S. Martin \cite{Ma}}) 
Consider a symplectic manifold $M$
acted on in a Hamiltonian fashion by $T = U(1)$ and 
 let $\mu$ denote the moment map for this action. Let $
t_0 < t_1$ be two regular values of $\mu$. 
Then we may subtract $t_i$ from the moment map to get a modified
symplectic quotient $\mtred(t_i)=\mu^{-1}(t_i)/T$ for $i=0,1$ with
maps
$$\Phi:H^{*}_{T}(M) \rightarrow H^{*}(\mtred(t_i))$$
sending $\eta$ to $(\eta)_{t_i}$. We then have
$$ \int_{\mtred(t_0) } (\eta e^{i \bom} )_{t_0} - 
\int_{\mtred(t_1) } (\eta e^{i \bom} )_{t_1} = 
n_0\sum_{F \in \calf: t_0 < \mu(F) < t_1} 
\!\!\!\!\!\!\!{\rm Res}_{X=0} e^{i \mu(F)( X) }
\int_F \frac{\eta(X) e^{i \omega}  }{e_F(X) } . $$
Here, $X \in \CC$  has been identified with $2 \pi i X \in \liet
 \otimes \CC$.



\end{prop}

\begin{corollary} \label{c:1}
 If $t_1 \in \liet$ is close enough to 
$t_0 \in \liet$ that there is a path between $t_0 $ and $t_1$ in 
$\mu(M)$ consisting entirely of regular values of $\mu$, then 
$$\int_{\mtred (t_0) } (\eta e^{i \bom} )_{t_0} = 
 \int_{\mtred(t_1) } (\eta e^{i \bom} )_{t_1}. $$
\end{corollary}

{\em Remark:} In fact both the equality
$$\int_{\mu_K^{-1}(0)/K} (\eta e^{i \bom} )_0 = \frac{(-1)^{n_+}}{|W|}
   \int_{\mu_T^{-1}(t)/T} (\cald^2 \eta e^{i \bom})_t$$
   of Proposition \ref{p:sm} and the residue formula of Proposition 
\ref{p:gkm} can
   be deduced easily from the residue formula of \cite{JK1} when $M$ is a
   compact symplectic manifold. However the proofs of Propositions \ref{p:sm}
   and \ref{p:gkm} can be adapted to apply in certain circumstances when
   $M$ is not compact and the residue formula of \cite{JK1}  is not valid.
   This will be crucial later.




\section{Extended moduli spaces}
To make progress we invoke a description of a 
symplectic space $\xc$ equipped with a Hamiltonian action of $K = SU(n)$ 
such that the symplectic quotient of $\xc$ at $0$ is 
$\mnd$, and
explain our general strategy
for obtaining (\ref{1.1}) and its
generalizations. The moduli space $\calm(n,d)$ was described by 
Atiyah and Bott \cite{AB} as the symplectic reduction
of an infinite dimensional symplectic vector space 
$\cala$
 with respect to 
the action of an infinite dimensional group $\calg$  (the 
{\em gauge group}).\footnote{To obtain his generating 
functionals, Witten formally applied his version of nonabelian 
localization to the infinite dimensional space $\cala$.}
 We however shall
exhibit $\mnd$ as the symplectic quotient of a {\em finite dimensional}
symplectic space $\xc$ by the Hamiltonian action of a finite
dimensional group $K$. In this case the group $K$ is $SU(n)$. 
One
characterization of the space $\xc$ is that it is the symplectic reduction 
of the infinite dimensional affine space $\cala$ 
by the action of the {\em based} gauge group $\calgo$ (which
is the kernel
of the evaluation map $\calg \to K$ at a prescribed basepoint: 
see
\cite{ext}).  Now if a compact group $G$ containing a 
closed normal subgroup
$H$  acts in a Hamiltonian fashion on a symplectic manifold $Y$, then 
one may \lq\lq reduce in stages'': the space $Y//H = \mu_H^{-1}(0)/H $ 
has a residual Hamiltonian action of the quotient group $G/H$ with
moment map $\mgh: Y//H  \to {(\lieg/\frak{h})}^* $, and 
$\mu_G^{-1}(0)/G$ is naturally identified as a symplectic 
manifold with 
$\mgh^{-1}(0)/(G/H). $ Similarly   $M(c)$ has a 
Hamiltonian action of $\calg/\calg_0 \cong K$, and the symplectic reduction 
with respect to this action is identified with the 
symplectic reduction of $\cala$ with respect to the full gauge group 
$\calg$.

A more concrete (and  entirely finite dimensional) characterization of 
$\xc$ is given 
in \cite{ext}.  
The space is defined by 
\begin{equation} \label{4.1} 
\mc = (\epsr \times \epc)^{-1} (\bigtriangleup) \subset \Hom (\FF, K) 
\times \liek. 
\end{equation}
Here, we 
identify $\homfk$ with 
$\ktg$ through a choice of generators $  \{x_1, \dots, x_{2g} \}$
for the free group $\FF$ on $2g$ generators; then 
 $\epsr: \Hom(\FF, K) \to K$ is the evaluation map on the relator
$r = \prod_{j = 1}^g [x_{2j-1}, x_{2j}]$ 
\begin{equation} \label{4.2} 
\epsr (h_1, \dots, h_{2g} ) = \prod_{j = 1}^g [h_{2j-1} , h_{2j} ]. 
\end{equation}
The map $\epc: \liek \to K $ is defined by 
\begin{equation} \label{4.3} \epc(Y) = \cent \exp (Y),   \end{equation}
where the generator $\cent$ of the 
centre of $K$ was defined at (\ref{1.p1}) above. 
The diagonal in $K \times K $ is denoted $\bigtriangleup$. 
The space $M(c)$ then has canonical projection maps 
$\proj_1, \proj_2 $ which make the following diagram commute: 
\begin{equation} \label{4.4} 
\begin{array}{lcr}
\xc  & \stackrel{\proj_2}{\lrar} & \liek \\
\scriptsize{\proj_1}
\downarrow & \phantom{\stackrel{aaaa}{\lrar} } & \downarrow \scriptsize{e_c} 
 \\
\homfk  & \stackrel{\epsr }{\lrar} & K\\ \end{array} \end{equation} 
In other words, $\xc$ is the fibre product of $\homfk $ and $\liek$ 
under the maps $\epsr$ and $\epc$. The action of $K$ on $\xc$ 
is given by the adjoint actions on $K$ and $\liek$. 
The space $\xc$ has the following properties (see 
\cite{ext} and \cite{J1}): 

\nc{\normcon}{\sigma}

\begin{prop} \label{p0}

{\bf
 (a)} The space $\xc$ is smooth near all $(h, \L) \in 
\homfk \times \liek$ for which the linear space 
$z(h) \cap \ker (d \exp)_\L \ne \{0 \} $. Here, $z(h) $ is the Lie
algebra of the stabilizer  $Z(h) $ of $h$. 




{\bf (b) } There is a $K$-invariant 2-form $\omega$ on 
$\homfk \times \liek$ whose restriction to $\xc$ is closed and which
defines a nondegenerate bilinear form on the 
Zariski tangent space to $\xc$ at  every $(h, \L)$ in an
open dense set in $\xc$ which includes the subset of $\xc$ where $\L=0$. 
(Thus the form $\omega$ gives rise to a symplectic structure
on an open dense subset of $\xc$.)



{\bf (c) } With respect to the symplectic structure given by the 
2-form $\omega$, a moment map $\mu: \xc\to \lieks$ for the action of 
$K$ on $\xc$ is given by 
$\normcon \proj_2$, where 
$\proj_2: \xc \to \liek$ is the projection map to 
$\liek$  (composed
with the canonical isomorphism $\liek \to \lieks$ given by the invariant 
inner product on $\liek$) and $\normcon = -2$. 

{\bf (d) } The space $\xc$
 is smooth in a neighbourhood of 
$\mu^{-1} (0).$ 

{\bf (e)} The symplectic quotient $\xred = \mu^{-1}(0)/K$ can be 
naturally identified with $\yyb/K = \mnd$.

\end{prop}

Using our description
of $\xc$ as a fibre product, 
it is easy to identify the components $F$ 
of the fixed point set of the action of $T$.
We examine the fixed point sets of the action of  $T$ on 
$\homfk$ and
$\liek$ and we find
\begin{equation} \label{4.5} 
\begin{array}{lcr}
\xc^T   & \stackrel{\proj_2}{\lrar} & \liet \\
\scriptsize{\proj_1}
\downarrow & \phantom{\stackrel{aaaa}{\lrar} } & \downarrow \scriptsize{e_c} 
 \\
{\rm Hom}(\FF, T)
  & \stackrel{\epsr }{\lrar} & 1 \in T \\ \end{array} \end{equation} 
(Notice that $\epsr$ sends ${\rm Hom}(\FF, T)$ to $1$ because 
$T$ is abelian.) Thus 
\begin{equation} \label{M^T} M(c)^T = {\rm Hom}(\FF, T) \times e_c^{-1}(1) = 
T^{2g} \times \{ \d - \tilde{c}: \onebl \d \in \intlat 
\subset \liet \} \end{equation}
where $\tilde{c} $ is a fixed element of $\liet $ for which 
$\exp \tilde{c} = c$. (Here, $\intlat$ denotes the 
integer lattice ${\rm Ker}(\exp) \subset \liet$.)
If we ignore the singularities of $M(c)$, this
description also enables us to identify the equivariant 
Euler class $e_{\fd}$ of the normal bundle of each component 
$T^{2g} \times (\d  - \tilde{c}) $ in $M(c)^T$
(indexed by $\d \in \intlat$).
This should be simply the equivariant
Euler class of the normal bundle to
$T^{2g}$  in $K^{2g}$, implying that 
$e_{\fd}$ is in fact independent of $\d$ and is given by 
\begin{equation} e_{\fd} (\xvec) = (-1)^{n_+g}
 \nusym(\xvec)^{2g}. \end{equation}
The symplectic volume of the component
$F_\d $ is independent of $\d$ (indeed these components are 
all identified symplectically with $T^{2g}$):
 we denote the volume of $F_\d$ 
by $\int_F \vol_\omega$.

The constant  value taken by the  moment map 
$\mu_T  $ on the component $F = F_\d$ is given by $\normcon(\d  - \ct)$
where $\normcon = -2$ as in Proposition 4.1(c).

We shall need also the following property (\cite{J2}):
\begin{prop}
The generating classes $a_r$, $b_r^j$ and $f_r$ $\!(r = 2, \dots, n$, 
$j = 1, \dots , 2g$) extend to classes $\tar$, $\tbrj$ and 
$\tfr$ $ \in H^*_K(\xc)$.  
\end{prop}
Indeed, the equivariant differential form $\tar \in \Omega^*_K(\xc)$
whose restriction represents the cohomology class $a_r \in H^*(\mnd)$
is simply the invariant polynomial $\tau_r \in S^r(\liek)^K$
$\cong \hk({\rm pt})$ which is associated to the $r$th Chern class.

Finally we shall need to work with the symplectic 
subspace $\emtc = \mu_K^{-1}(\liet)$ 
of $\xc$, which is no longer acted on by $K$ but is acted on by $T$. 
The space $\emtc$ has an important periodicity property:

\begin{lemma} \label{l4.3} 
Suppose $\tran \in {\rm Ker} (\exp ) \subset \liet$. 
Then
there is a homeomorphism $s_\tran: \emtc \to \emtc$ defined by 
$$ s_\tran: (h, \L) \mapsto (h, \L + \tran). $$
\end{lemma}


\section{Proof of the residue formula}
Na\"{\i}ve application of the residue formula (Theorem 
\ref{t4.1}) to the
space $M(c)$, ignoring the fact that it is noncompact and has
singularities,
would   thus yield
\begin{eqnarray} \label{1.7}
\prod_{r = 2}^n a_r^{m_r} \exp ({f_2}) [\mnd] & = & \cr
 & & \hskip-4cm n_0 C^K
{\rm Res}
 \Biggl (  \nusym^2(X) (\int_F \vol_\omega)  \sum_{\d \in \intlat} 
\frac{ \prod_{r = 2}^n \tau_r(X)^{m_r}e^{\isq (-\ct + \d) (X)}  }
{(-1)^{n_+ g} \nusym^{2g}(X)} \Biggr ). \end{eqnarray}
The main problem with (\ref{1.7}) (related to the noncompactness of $\xc$,
which permits the fixed point set $M(c)^T$ to consist of infinitely 
many components $F_\d$) is that the sum over $\d$
does not converge for $\xvec \in \liet$. In this section we shall sketch
a sequence of results that enable us nonetheless to conclude
that (\ref{1.7}) is true if interpreted appropriately.
We shall concentrate mainly on the case when $n$ is $2$ but
the argument can be generalized to higher $n$; details
will be given in a subsequent paper. (See Theorem 
\ref{t11} and the remarks following it.)



\begin{lemma} \label{l1}
For generic $\xi \in \liet$, 
\begin{eqnarray} \mexp & \eqdef &
\Bigl \{   (h_1, \dots, h_{2g}, \L )  \in K^{2g} \times 
\liek \; :\nonumber\cr
\; & & \hskip1em\prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} 
= c \exp (\xi) \exp (\L) \Bigr \} \end{eqnarray}
is a smooth manifold.
\end{lemma}

\Proof We see that $\mc = F^{-1}(c)$ where 
$F: K^{2g} \times \liek \to K$ is defined by 
$$
 F \Bigl ( h_1, \dots, h_{2g}, \L \Bigr )  
= \prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} 
\exp (-\L). $$
By Sard's theorem, for generic $\xi \in \liek$, 
$c \exp (\xi)$ is a regular value of $F$ (although
$c$ itself cannot be a regular value of $F$ since
$\mc$ is not smooth, see \cite{J1},\cite{J2}). Moreover, since $F$ is 
$K$-equivariant, we may assume without loss of generality that $\xi \in \liet$.
$\square$ 



\noindent{\em Remark:}
 Note that when $\exp (\xi)=1$ we recover the definition of $M(c)$.




\begin{lemma} \label{l2}
Define $\mtcexp) = \mu^{-1} (\liet) 
\cap \mexp$, where $\mu: K^{2g} \times \liek \to \liek$ is defined by 
$$ \mu (   h_1, \dots, h_{2g}, \L )    = \normcon \L$$
(see Proposition \ref{p0}(c))
and $\mu_T$ is the projection  of $\mu$ onto $\liet$.
Then there is a $T$-equivariant homeomorphism 
$s_{\xi}: \mtcexp \to \emtc$ which sends 
$\mu_T^{-1}(0) \cap \mtcexp$ 
to $\mu_T^{-1}(\normcon \xi) \cap \emtc$. 
\end{lemma}
\Proof One may define the homeomorphism as follows:
$$ s_\xi:
 (h_1, \dots, h_{2g},\L ) \to (h_1, \dots, h_{2g},\L +
\xi). \bla \square$$
\nc{\sx}{s_\xi^*}
\begin{lemma} \label{l3} 
We have 
$$ \int_{\xred} \Phi(\eta e^{i \bom} ) = 
\frac{1}{|W|}\int_N \sx \Phi(\cald \eta e^{i \bom})$$
where 
$$N=\mu_T^{-1}(0) \cap \mtcexp/T $$
and $|W| = n!$ is the order of the Weyl group of 
$K = SU(n)$. 
\end{lemma}
\Proof We first identify
$\int_{\xred}\Phi(\eta e^{i \bom} ) $ with 
$$ \frac{1}{|W|} \int_{\mu^{-1}(0)/T} 
\Phi(\nusym\eta e^{i \bom} )
= \frac{1}{|W|} \int_{\mu_T|_{\emtc}^{-1}(0)/T } 
\Phi(\nusym\eta e^{i \bom} ), $$
via Proposition \ref{p:sm}, whose proof can be made to work
in this situation, even though $M(c)$ is noncompact and singular,
because $\mu^{-1}(0)$ is compact and $M(c)$ is nonsingular in a 
neighbourhood of $\mu^{-1}(0)$. 
Then we observe that for 
sufficiently small $\xi \in \liet$, 
$$\int_{\mu_T|_{\emtc}^{-1}(0)/T } 
\Phi(\nusym\eta e^{i \bom} ) = 
\int_{\mu_T|_{\emtc}^{-1}(\normcon \xi)/T } \Phi (\nusym\eta e^{i \bom} ) $$
(by Corollary \ref{c:1}). Finally we use the
homeomorphism $s_{\xi}$ of Lemma \ref{l2}
to establish an identification between
$\mu_T|_{\emtc}^{-1}(\normcon \xi)/T$ and $ N =$ 
${\mu_T|_{\mtcexp}^{-1}(0)} /T. \square$ 


Let us now examine the behaviour of the images in $\hht(\emtc)$ 
of the generating classes
$\tar, \tbrj, \tfr$ $ \in \hk(\mc)$ under pullback under the 
homeomorphisms $s_{\L_0}: \emtc \to \emtc$
defined at Lemma \ref{l4.3}. By abuse of language, we shall
refer to these images also as $\tar, \tbrj $ and $\tfr$. 
It follows from \cite{J2} that the classes $\tar$ are the images in
$\hk(\mc)$ of
the polynomials $\tau_r \in \hk = S(\lieks)^K$. Moreover 
(\cite{J2}, (8.18)) the classes
$\tbrj \in \hk(\mc)$ are of the form 
$\tbrj = \proj_1^* (\tbrj)_1 $ where $  (\tbrj)_1 \in \hk(K^{2g} )$ 
and $\proj_1: \mc \to K^{2g}$ is the projection in
(\ref{4.4}).   It follows that
$\stran^* \tbrj = \tbrj$ and 
$\stran^* \tar = \tar. $ 

Furthermore we see from (8.30) of \cite{J2} that $\tf(X)$ is of 
the form 
\begin{equation} \label{8.3} \tf(X) = \proj_1^* f_2^1 + \langle \mu,X \rangle \end{equation}
where $f_2^1 \in H^*_K(K^{2g})$ and
 $\mu: \mc \to \liek $ is the moment map 
(which is $\normcon$ times the restriction to $\mc$ of  the
projection $K^{2g} \times \liek \to \liek$: see 
Proposition \ref{p0}(c)). 
 It follows from the definition (\ref{8.3}) of $\tf$ that
for any $\tran $ in the integer lattice of $\liet$ (the kernel
of the exponential map),  
\begin{equation} \label{8.4} \stran^* \tf(X) = \tf(X) + \normcon
\langle \tran,X \rangle . \end{equation}
Note also that there is a commutative diagram of
homeomorphisms
\begin{equation}
\begin{array}{lcr}
\mtcexp   & \stackrel{s_\xi}{\lrar} & \mtc \\
\scriptsize{s_{\tran} }
\downarrow & \phantom{\stackrel{aaaa}{\lrar} } & \downarrow 
\scriptsize{s_{\tran} } 
 \\
\mtcexp
  & \stackrel{s_\xi }{\lrar} & \mtc \\ \end{array} 
\end{equation}

For simplicity, we restrict from now on to $K = SU(2)$, and let 
$\tran =$\newline $ 2 \pi im  {\rm diag}(1,-1)$ (for $m \in \ZZ$) and 
identify $X \in \RR$ with $2 \pi i X {\rm diag}(1,-1) \in \liet$. 




\nc{\tef}{e_F^{(N)} }

\nc{\trivfac}{2}
\begin{lemma} \label{l5.4} Suppose $\eta$ is a polynomial in the  $\tar$.
Then for any positive integer $m$ we have that 
$$ \int_N \sx \Phi(
\eta e^{i \bom} e^{ im \normcon X}) =
\int_{\mtcexp \cap\mtinv ( \normcon m)/T}\sx \Phi( \eta e^{i \bom}) 
$$
$$ =\int_N \sx \Phi(\eta e^{i \bom} ) +
\trivfac \, {\rm sgn}(\normcon) 
 \sum_{F \in \calf, 0 < |\mu_T(F)| <  |\normcon| m} 
{\rm Res}_{X = 0 } \Bigl (   \int_F 
\frac{\eta(X) e^{i \bom(X)} \g(X)} {e_F(X) } \Bigr ). $$
Here,  $\calf$ is the set of connected components of the fixed point set of
the action of $T$ on $M(c)$ and  if $F\in\calf$ then $e_F$
denotes the equivariant Euler class of the normal 
bundle of $s_{\xi}^{-1}(F)$ in $M(c \exp \xi)$. Also $\g(X)=2X$ denotes
the positive root of $SU(2)$ and $\normcon= -2$ is the  normalization constant
resulting from the  normalization of the symplectic
form (see Proposition 
\ref{p0} (c)). The overall factor of $\trivfac$ multiplying the sum over $\calf$ 
results from the fact that the element $ - 1 \in U(1) $ acts trivially 
on $\mc$. 

\end{lemma}
\Proof We see that 
$$ \int_{\mtcexp \cap\mtinv (\normcon m)/T} 
\sx \Phi( \eta e^{i \bom} )
=  \int_N \sx \Phi(\stranm^* \eta e^{i \bom} )
$$
$$ \onebl 
=  \int_N \sx \Phi( \eta e^{i \bom} e^{ i m\normcon X} ) $$
by  (\ref{8.4}).  
By (\ref{M^T}) the set of components of the fixed point set of the action
of $T$ on $M(c \exp \xi)$  is $\{s_{\xi}^{-1}(F):F\in\calf\}$. Using the fact 
that the  root $-\g(X)=-2X$ of $SU(2)$ represents the
Poincar\'e dual of $M_{\liet}(c \exp \xi)$ in $M(c \exp \xi)$, Guillemin and
Kalkman's proof of Proposition \ref{p:gkm}
can be applied to compactly  supported equivariant forms 
in a neighbourhood
of $\mtcexp \cap \mu^{-1}_T[\normcon m, 0]$ in the smooth
manifold $M(c \exp\xi)$ to show that
$$- {\rm sgn}(\normcon) \Bigl ( 
  \int_{\mtcexp \cap\mtinv ( \normcon m)/T}\sx \Phi
(\eta e^{i \bom})
-  \int_N\sx \Phi( \eta e^{i \bom} ) \Bigr ) 
$$ 
$$ \phantom{aaaa} = - \trivfac  
\sum_{F \in \calf, 0 < |\mu_T(F)| <  |\normcon| m} 
{\rm Res}_{X = 0 } \Bigl (  \int_F 
\frac{\eta(X) e^{i \bom(X)} \g(X)} {e_F(X) }\Bigr ). \square $$


\begin{corollary} \label{l5} 
For all $j \ge 0$,
$$-{\rm sgn}(\normcon)\int_N s_\xi^*\ \Phi  (X^j (e^{ i m \normcon  X} - 1)
 \eta e^{i \bom} ) 
= \frac{(-1)^{g-1}}{2^{2g-2} }\sum_{k = 1}^m {\rm Res}_{X = 0} 
 \frac{e^{i \normcon ( k  - 1/2 ) X }
\int_F  \eta e^{i \omega}  }{X^{2g-1-j} }. $$
\end{corollary}
\Proof This follows from Lemma  \ref{l5.4} using the fact that 
$e_F(X) = (-1)^g 
\gamma(X)^{2g}$ where $\gamma(X) = 2X$ is the positive root of 
$SU(2)$.$\square$

We shall encode this information by defining a
formal Laurent series
$L_\eta$ in an indeterminate $x$ as follows: 
\begin{equation} \label{8.5}
L_\eta(x) = \sum_{j \ge 0} 
\frac{\int_N s_\xi^* \Phi  (\eta X^j e^{i \bom} )}{x^{j+1}}. \end{equation}
We appeal to the following elementary observation:
\begin{lemma} \label{l6}Suppose $f(x)$ $ =
\sum_{j \in \ZZ} f_j x^j $ is a formal Laurent series
in $x$. Then
$$ \sum_{j \ge 0} \frac{ {\rm Res}_{X=0} (X^j f(X)) }{x^{j+1} } 
= \calp( f(x)), $$
where $\calp$ denotes the principal part. 
\end{lemma}

We have also
\begin{prop} \label{p7}
For any formal power series $g(x)$ $ = \sum_{j \ge 0 } g_j x^j$ 
in $x$, 
$$ \calp(L_{g(X) \eta}(x) )= \calp( g(x) L_\eta(x)). $$
\end{prop}
\Proof This follows because for any $k \ge 0$, 
$$L_{X^k \eta} (x)  
= \sum_{j \ge 0 } \frac{ 
\int_N \sx \Phi(\eta X^{k + j} ) e^{i \bom} }
{x^{k + j+1} } x^k. $$

\begin{prop} \label{p8}
$$-{\rm sgn}(\normcon) \calp( (e^{ i \normcon x  } - 1 ) L_\eta ) = 
\frac{(-1)^{g-1}}{2^{2g-2}}
\calp\frac{ (\int_{T^{2g}} \eta e^{i \omega} ) e^{i \normcon x/2} }
 {x^{2g-1} }. $$
\end{prop}
\Proof
$$ 
\calp\Bigl ( (e^{ i \normcon x}  - 1 ) L_\eta(x) \Bigr ) 
= \calp\Bigl ( L_{(e^{ i \normcon X}  - 1 ) \eta} (x)  \Bigr )
\;\;\;  \mbox{by Proposition 
\ref{p7} }$$
$$ = \calp\Bigl ( \sum_{j \ge 0} 
x^{-j-1} \int_N \sx \Phi \Bigl (  (X^j \eta e^{i \bom} ) 
(e^{ i \normcon X} - 1 ) \Bigr ) \;\; \mbox{by definition of $L_\eta$} $$
so that 
$$ - {\rm sgn}(\normcon) 
\calp\Bigl ( (e^{ i \normcon x}  - 1 ) L_\eta(x) \Bigr ) 
=\frac{(-1)^{g-1}}{2^{2g-2}}  
\calp \sum_{j \ge 0} 
x^{-j-1} {\rm Res}_{X=0} 
\Bigl ( \frac{ ( \int_{T^{2g}} \eta(X)  e^{i \omega} )e^{i \normcon X/2} } 
{X^{2g-1-j} } \Bigr ) $$
(by Corollary \ref{l5} when $m  = 1$ )
$$ = \frac{(-1)^{g-1}}{2^{2g-2} }   \calp \; \frac{ (\int_{T^{2g} } \eta(x) 
 e^{i \omega} ) e^{i \normcon x/2}  }
  {x^{2g-1} } \;\;\mbox{by Lemma \ref{l6}. }\square $$

\begin{corollary} \label{p9}
$$ -{\rm sgn}(\normcon) L_\eta (x)
= \frac{(-1)^{g-1}}{2^{2g-2} }
 \frac{ \int_{T^{2g} } \eta e^{i \omega} e^{i \normcon x/2} }
{ (e^{ i \normcon x} - 1 ) x^{2g-1} } + h(x) $$
where $h $ is a formal Laurent series 
with at most a simple pole at $0$.
\end{corollary}

\Proof This follows immediately when we divide through
the result of Proposition \ref{p8} by 
$(e^{ i \normcon x} - 1 )$. $\square$



Unwinding the definition of $L_\eta$ we have 
 from Corollary \ref{p9} 
 (provided that $k   \ge 1  $) that  
$$ \int_N s_\xi^* \Phi (X^k\eta e^{i \bom} )  = {\rm Res}_{x = 0 } 
\Bigl ( x^{k} L_\eta(x) \Bigr )$$
\begin{equation} \label{8.10}
= \frac{(-1)^g}{2^{2g-2} } 
{\rm Res}_{x=0} \frac{1}{2i   x^{2g-1-k}\sin (|\normcon| x/2) }  ( \int_{T^{2g} }
 \eta(x)  e^{i \omega} ). \end{equation}
It follows finally that 
\begin{theorem} \label{t11}
$$\int_{M(2,1)} a_2^j e^{i f_2} = 
\frac{(-1)^g}{2^{2g-1}  i } \int_{T^{2g}} e^{i \omega} 
 {\rm Res}_{X=0} \Bigl (\frac{1} {  X^{2g-2 - 2j} \sin 
|\normcon| X/2 } \Bigr ), $$
where the normalization is such that the class $a_2 = \Phi( X^2) $.
\end{theorem}





\Proof This is true since 
$$ \int_{M(2,1)} a_2^j e^{i f_2}   = \int_{M(2,1)} 
\Phi \Bigl (   X^{2j} e^{i \bom} \Bigr )  $$
(by the remarks in the paragraph above (\ref{8.3}), and 
noting that  $\tau_2 (X) =  X^2$)
$$  =  \int_N  s_\xi^*
\Phi (X^{2j+1} e^{i \bom} )  \;\; \mbox{(by Lemma \ref{l3}).}$$
(Notice that $\nusym(X) = \g(X) =2X$ in our
chosen normalization.)
We then combine this fact with (\ref{8.10}). 
 $\square$ 

A short calculation using the results of \cite{J2} (particularly
the description of the symplectic form using the formulas
(7.11)-(7.12) for the fundamental class) shows that
$$\int_{T^2} \omega = -2. $$ 
This gives $vol_\omega (T^{2g} ) = (-1)^g 2^g. $ So, recalling from
Proposition 4.1(c) that $\s = -2$, we find finally 
\begin{equation} \label{5.006} 
\int_{M(2,1)} a_2^j e^{i f_2} = \frac{ i^{g-1} } {2^{g-1} } 
 {\rm Res}_{X=0} \Bigl ( \frac{1}{  X^{2g-2 - 2j} \sin 
 X } \Bigr ). \end{equation}
Thus we have recovered 
Witten's formula (see 
(\ref{5.002})  below): this result was first obtained by 
Donaldson \cite{Do} and Thaddeus \cite{T}. 
\begin{theorem} The intersection numbers of the classes 
$a_2$ and $f_2$ on $M(2,1)$ are given by 
\begin{equation} \label{5.007}
\int_{M(2,1)} a_2^j e^{ f_2} = \frac{(-1)^{g-1-j}  } {2^{g-1} } 
 {\rm Res}_{X=0} \Bigl ( \frac{1}{  X^{2g-2 - 2j} \sin 
 X } \Bigr ). \end{equation} 
\end{theorem}








\noindent{\em Remark:} Notice that formally we can express the result
of Theorem \ref{t11} as
$$\int_{\calm(2,1)} a_2^j e^{i f_2} 
 =   (-1)^g {\rm Res}_{X = 0 } 
\Bigl ( \sum_{k \geq 0}  \frac{ (2X)^2 X^{2j} e^{(2k+1)  i \normcon X/2}
\int_{T^{2g}} e^{i \omega} }
{ (2X)^{2g} } \Bigr ), $$
which is what we would expect from applying the residue formula of 
\cite{JK1} formally to $\mc$. However this sum does not converge
in a neighbourhood of $0$ and the sum of the residues at $0$ does
not converge. 





Witten's result (using (4.29) and (4.44) of 
\cite{tdgr}) may be expressed in the following form for 
$j \le g-2$: 
\begin{equation} \label{5.001} 
\int_{M(2,1)} \Theta^j e^{\omega}  = \frac{(-1)^{g-1-j}}{2^{g-2}
\pi^{2(g-1-j)} } \sum_{n = 1}^\infty \frac{ (-1)^{n+1}}{n^{2(g-1-j)} }
\end{equation} 
(taking into account 
orientations and the fact that $M(2,1) $ is a $2^{2g}$-fold cover
of the moduli space Witten studies). 
To show this agrees with our result (\ref{5.007}),
we use the following
fact
 which can be established by an elementary contour integral argument:
\begin{lemma} \label{l11} For any positive integer $m$, we have
$$\sum_{n > 0} \frac{(-1)^{n+1} }{n^{2m} } = 
 \pi i  {\rm Res}_{X = 0} \frac{e^{i \pi X} } {X^{2m} ( e^{2 i \pi X}-1) }
= \frac{\pi^{2m} }{2} {\rm Res}_{Y= 0} \frac{1}{ Y^{2m} \sin Y}. $$
\end{lemma} 


Using Lemma \ref{l11}, (\ref{5.001}) becomes 
\begin{equation} \label{5.002} \int_{M(2,1)} \Theta^j e^{\omega}
= \frac{(-1)^{g-1-j} }{2^{g-1}} {\rm Res}_{Y = 0 } 
\frac{1}{Y^{2(g-1-j) } ( \sin Y )} , \end{equation}
in agreement with (\ref{5.007}). 

The sum in (\ref{5.001}) may be expressed in terms of 
zeta functions through  the following fact which follows from 
an easy computation:
\begin{lemma} \label{l12} For any positive integer $m$, we have
$$\sum_{n > 0} \frac{(-1)^{n+1} }{n^{2m} } = 
 (1 - \frac{1}{2^{2m-1} } ) \zeta(2m) . $$
\end{lemma}

\noindent{\em Remark:} The proof of Theorem \ref{t11}
given above can be generalized to 
give formulas for intersection pairings involving the $b_2^j$ as well
as $a_2 $ and $f_2$. These were already treated in Thaddeus' work
\cite{T}.

\noindent{\em Remark:}
Lemma \ref{l11} establishes directly that for rank $2$ and degree $1$
the formula (\ref{1.3}) obtained by Witten (expressing
intersection numbers in terms of a sum over irreducible representations
of $SU(2)$) may be rewritten as the residue that appears in our
proof.
We can generalize this argument to cover $n$ higher than 
$2$, getting formulas involving
iterated $1$-variable residues similar to the residue in 
Theorem \ref{t11}. Details will appear in a later paper \cite{JK2}.
 For general coprime $n$ and $d$ the
work of Szenes \cite{Sz} establishes the equivalence of the 
sum (\ref{1.1}) obtained by Witten and the residue 
(\ref{1.7}) which appears in our proof. 
Szenes' proof  is a multidimensional generalization of the contour integral 
argument that establishes Lemma \ref{l11}.















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\end{document}