Heat kernels, symplectic geometry, moduli spaces and finite groups (Q2739544)

One of the important roles played by heat kernels is the celebrated index theorem which relates analytic information with topological information by local (as \(t \to 0\)) and global calculation. For a map \(f: M \to N\) between two spaces, a given heat kernel can be pulled-back or pushed-forward in order to obtain the heat kernel for the unknown side and one can obtain topology and geometry of the unknown side. Then local (for \(t\to 0)\) and global calculations can be done. The paper under review applies this principle to (i) symplectic reductions, (ii) moduli spaces of stable parabolic bundles over a Riemann surface, (iii) finite groups and (iv) pushed-forward measures. The method unifies several results treated by Freed-Quinn, Jeffrey-Kirwan, Witten and Wu. NEWLINENEWLINENEWLINEIn section 2, the principle is well-explained. Let \(\mu: M \to {\mathfrak g}^*\) be a moment map with respect to a symplectic \(G\)-action. For any semisimple compact Lie group \(G\), one has the standard heat kernel \(H(t, x, x_0)= \frac{1}{(4\pi t)^{n/2}}\exp\left(- \frac{\|x-x_0\|^2}{4t}\right)\) over \({\mathfrak g}^* \cong\mathbb{R}^n\) (as a vector space). The pull-back \(\mu^*(H(t, x, 0))\) on \(\mu^{-1}(0)\) against the symplectic volume form \(e^{\omega}\) on \(M\) gives NEWLINE\[NEWLINEI(t) = \int_M H(t, \mu(y), 0)e^{\omega}.NEWLINE\]NEWLINE (i) Local calculation for \(t \to 0\): \(I(t)\) localizes to a neighborhood of \(\mu^{-1}(0)\) in \(M\). If the neighborhood can be identified with \(\mu^{-1}(0) \times B_{\delta}\) for a small \(\delta\)-ball in \({\mathfrak g}^*\), then NEWLINE\[NEWLINEI(t) = |G|\int_{M_G}e^{\omega_0 - t \langle F, F\rangle} + o(e^{-\delta^2/4t}),NEWLINE\]NEWLINE where \(\omega = \pi^* \omega_0 + d(\alpha, \theta), F = d \theta - \theta \wedge \theta\) and \(\pi: \mu^{-1}(0) \to \mu^{-1}(0)/G = M_G\) provided 0 is a regular value of the moment map (for singular value, \(M_G\) is a singular stratified symplectic space, some calculations can be extended);NEWLINENEWLINENEWLINE(ii) Global calculation uses the equivariant cohomology class. It leads to the Witten identity NEWLINE\[NEWLINE|G|\int_{M_G}e^{\omega_0} = \lim_{t \to 0}\int_{\mathfrak g} e^{-t\langle \varphi, \varphi\rangle} \int_Me^{\omega + i (\mu , \varphi)} d \varphi.NEWLINE\]NEWLINE This entails a circle of ideas due to Jeffrey-Kirwan, Martin, Witten and Wu. NEWLINENEWLINENEWLINEBy applying the same principle to \(M = G^{2g} \times G^n\), \(N = G\) and \(f=\prod_{i=1}^g[x_i, y_i] \prod_{j=1}^n z_j c_j z_j^{-1}\) and the heat kernel on \(G\), the space \(M_c = f^{-1}(e)/G\) is the moduli space of \(G\)-flat connections with prescribed holonomies along punctured circles in a Riemann surface. The symplectic volume of \(M_c\) can be calculated in section 3. Similarly, applying this to the finite group case gives the counting of the number of solutions in section 5. In section 4, the pushed-forward measures for the map \(f\) studied in section 3 and for the \(n\)th commutators are given.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00007].