The generating function for certain cohomology intersection pairings of the moduli space of flat connections (Q2780733)

Though all principal \(SU(2)\)-bundles over closed Riemann surfaces are topologically trivial, they give rise to rich moduli spaces of flat connections. Such moduli spaces come up in a variety of settings, including mathematical physics (conformal field theory), symplectic geometry, and the representation theory of surface groups. In these settings it is also interesting to consider flat connections with prescribed holonomy around a set of fixed points on the surface. The moduli spaces studied by the author are of this type. Specifically, for a closed oriented surface \(\Sigma^g\) of genus \(g\) with marked points \(\{p_1,\dots,p_n\}\), he considers moduli spaces \(\mathcal{M}_g(t_1,\dots,t_n)\) of flat \(SU(2)\) connections with holonomy around \(p_i\) specified by \(t_i\). NEWLINENEWLINENEWLINEThe main results in this paper concern the cohomology ring for \(\mathcal{M}_g(t_1,\dots,t_n)\). The author uses certain naturally defined cohomology classes to probe the ring structure. The classes, denoted \(r^g_1,\dots,r^g_n\), arise as first Chern classes of natural \(S^1\)-bundles over \(\mathcal{M}_g(t_1,\dots,t_n)\). In addition, the moduli space has a natural symplectic structure, the symplectic form for which defines a cohomology class. Denoting the symplectic form by \(\omega^g_{t_1,\dots,t_n}\), the author computes a generating function for intersection pairing of the form NEWLINE\[NEWLINE\int_{\mathcal{M}_g(t_1,\dots,t_n)}(r^g_1)^{k_1}\dots (r^g_n)^{k_n}(\omega^g_{t_1,\dots,t_n})^{3g+n-3-k},NEWLINE\]NEWLINE where \(k=\sum k_i\) and \(3g+n-3\) is the dimension of \(\mathcal{M}_g(t_1,\dots,t_n)\). NEWLINENEWLINENEWLINEThe methods used come from symplectic geometry. The moduli spaces as well as the \(S^1\)-bundles over them can be described as symplectic quotients. This allows one to use the Duistermaat-Heckman theorem, from which one obtains a relation between the desired intersection pairings and symplectic volumes for \(\mathcal{M}_g(t_1,\dots,t_n)\). The volume formula of \textit{L. Jeffrey} and \textit{J. Weitsman} [Math. Ann. 307, 93-108 (1997; Zbl 0911.58017)] can then be invoked, leading to the final formula.NEWLINENEWLINENEWLINETheorem: For \(x_1,\dots,x_n\in \mathbb{R}\) NEWLINE\[NEWLINE\begin{aligned} \sum &\frac{x_1^{k_1}}{k_1!}\dots\frac{x_n^{k_n}}{k_n!} \int_{\mathcal{M}_g(t_1,\dots,t_n)}(r^g_1)^{k_1}\dots (r^g_n)^{k_n} \frac{(\omega^g_{t_1,\dots,t_n})^{3g+n-3-k}}{(3g-3+n-k)!}\\ &=\frac{1}{2^{g-2}\pi^{2g-2+n}}\sum_{m=1}^{\infty} \frac{\prod_{j=1}^{n}\sin(\pi m(t_j+x_j))}{m^{2g-2+n}} \end{aligned}NEWLINE\]NEWLINE where the sum is over all positive \(k_1,\dots,k_n\) such that \(k=\sum_{j=1}^n k_j \leq 3g-3+n\).NEWLINENEWLINENEWLINEAs corollaries, the author obtains (1) an explicit formula involving Bernoulli polynomials in the case \(n=1\), and (2) a proof that, when \(n=1\), the class \((r_1^g)^{2g-1}\) does not vanish. This last result complements \textit{J. Weitsman}'s result [Topology 37, 115-132 (1998; Zbl 0919.14018)] that \((r_1^g)^k=0\) for \(k\geq 2g\).