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Geometry Seminar

Tyurin data, Cauchy kernels, and Goldman Poisson structure (SPECIAL TIME noon-1pm)
Friday, March 5, 2021
12:00-1:00 PM
Off Campus Location
In this talk I will overview constructions and results of a joint work with Marco Bertola and Chaya Norton ( https://arxiv.org/abs/2102.09520). We consider the moduli space M of vector bundles of rank r and degree rg over a fixed (smooth and compact) Riemann surface of genus g. These admit a convenient explicit parametrization due to Tyurin, which we re-derive under general assumptions; this provides complex coordinates on M. In terms of these coordinates we explicitly describe the "theta divisor" in M of bundles with h^1>0 (whereas generic bundles in M have h^1=0). On the complement of this divisor we introduce the notion of matrix Cauchy kernel; this is a generalization of a classical construction in the case of line bundles. The Cauchy kernel depends holomorphically on the moduli, and for each bundle it is a differential of one variable and a function of the other on the base Riemann surface; it has a pole along the diagonal with identity residue and the subleading regular term provides an affine holomorphic (hence flat) connection, which we term "Fay connection". The holonomy of this connection thus gives a holomorphic map of the complement M_0 of the theta divisor in M into the GL(r)-character variety. Even more, the cotangent bundle T M_0 is well-known to be identified with Higgs fields, and by adding any Higgs field to the Fay connection one obtains new connections; by taking holonomy of these connections we map the whole TM_0 into the GL(r)-character variety. Our main result is the identification of the Goldman symplectic structure on the GL(r)-character variety with the canonical symplectic structure on T*M under such holonomy map; the proof exploits these constructions, also inspired by works of Krichever, as well as a prior work by Alekseev and Malkin and a recent one by Bertola and Korotkin. We finally discuss the behavior of the canonical symplectic potential, considered as a one-form in the character variety via the holonomy map; it has a simple pole on the image of the theta-divisor in the character variety, with residue equal to h^1. Speaker(s): Giulio Ruzza (Université catholique de Louvain)
Building: Off Campus Location
Location: Virtual
Event Type: Workshop / Seminar
Tags: Mathematics
Source: Happening @ Michigan from Department of Mathematics