The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

TL;DR

This paper investigates the geometric structure of moduli spaces of parabolic connections with quadratic differentials, revealing their symplectic and twisted cotangent bundle structures, and linking them to isomonodromic deformation systems.

Contribution

It introduces a symplectic structure on these moduli spaces and demonstrates their interpretation as twisted cotangent bundles, advancing understanding of their geometric and deformation properties.

Findings

Moduli spaces are endowed with symplectic structures.

They are shown to have twisted cotangent bundle structures.

Connections to isomonodromic deformation systems are established.

Abstract

In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.