Periods of meromorphic quadratic differentials and Goldman bracket

TL;DR

This paper explores the symplectic structure of monodromy maps for second-order linear equations with meromorphic potentials, extending known results from holomorphic to meromorphic cases with simple poles on Riemann surfaces.

Contribution

It demonstrates that the Goldman bracket on the character variety arises from the canonical symplectic structure on the cotangent bundle in the meromorphic setting, extending previous holomorphic results.

Findings

Goldman bracket corresponds to the symplectic structure for meromorphic potentials

Extension of symplectic properties from holomorphic to meromorphic cases

Monodromy map preserves symplectic structure in the meromorphic setting

Abstract

We study symplectic properties of monodromy map for second order linear equation with meromorphic potential having only simple poles on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*M_{g,n}$ implies the Goldman bracket on the corresponding character variety under the monodromy map, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials with simple poles.