Gromov-Witten invariants of flag manifolds and products of conjugacy classes (Q2752201)

Let \(K\) be a compact, connected, simply-connected Lie group. Denote by \(\Delta\) the set of \((n+1)\)-tuples \((\lambda_{0},\lambda_{1},\ldots, \lambda_{n})\) in a Weyl alcove, such that the product of the conjugacy classes \(C_{\lambda_{0}}, C_{\lambda_{1}}, \ldots, C_{\lambda_{n}}\) of \(K\) contains the unit. In [\textit{E. Meinrenken} and \textit{C. Woodward}, J. Differ. Geom. 50, 417-469 (1998; Zbl 0949.37031)], the authors prove that \(\Delta\) is a convex polytope. The polytope may be determined in terms of the Gromov-Witten invariants of \(G/P\), where \(G\) is the complexification of \(K\) and \(P\) is a maximal parabolic subgroup. NEWLINENEWLINENEWLINEAn application to the moduli space of flat connections is given. Let \(\Sigma\) be a surface with boundary \(\partial \Sigma = (S^{1})^{n+1}\) and let \(R_{K}(\Sigma)\) be the space of flat connections on the trivial \(K\)-bundle modulo the gauge transformations which are the identity on \(\partial\Sigma\). Then, if \(LK\) denotes the loop group of \(K\), there is a Hamiltonian action of \(LK^{n+1}\) on \(R_{K}(\Sigma)\). The associated convex polytope \(\Delta\) parametrizes the possible holonomies around boundary components which give a non-empty moduli space of flat connections. NEWLINENEWLINENEWLINEA formula for the volumes of the spaces \(R_{K}(\Sigma,\lambda)\) of flat connections with fixed holonomies is also given.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].