An Interpretation of Some Hitchin Hamiltonians In Terms of Isomonodromic Deformation

TL;DR

This paper interprets certain Hitchin Hamiltonians as Hamiltonian vector fields on moduli spaces of irregular connections, linking isomonodromic flows with hypercohomology and symplectic structures on these spaces.

Contribution

It adapts previous work to interpret Hitchin Hamiltonians in terms of isomonodromic deformations on moduli spaces of irregular connections.

Findings

Symplectic structure from Atiyah--Bott form matches hypercohomology description.

Hitchin Hamiltonians correspond to isomonodromic flow differences.

Moduli space local structure via Hecke modifications.

Abstract

This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott form agrees with the one given in terms of hypercohomology. The main results of this paper adapt work of Krichever and of Hurtubise to give an interpretation of some Hitchin Hamiltonians as yielding Hamiltonian vector fields on moduli spaces of irregular connections that arise from differences of isomonodromic flows defined in two different ways. This relies on a realization of open sets in the moduli space of bundles as arising via Hecke modification of a fixed bundle.