Harmonic Maps and Hypersymplectic Geometry

TL;DR

This paper explores the hypersymplectic geometry of a moduli space related to Hitchin's harmonic map equations, focusing on solutions with small Higgs fields and their geometric properties in split-signature settings.

Contribution

It introduces a smooth open subset of the moduli space where hypersymplectic geometry can be studied despite global indefiniteness issues.

Findings

Constructed a smooth open set of solutions with small Higgs fields.

Analyzed the hypersymplectic structure in split-signature moduli space.

Reinterpreted results in terms of Riemannian geometry of G-connection moduli space.

Abstract

We study the hypersymplectic geometry of the moduli space of solutions to Hitchin's harmonic map equations on a $G$-bundle. This is the split-signature analogue of Hitchin's Higgs bundle moduli space. Due to the lack of definiteness, this moduli space is globally not well-behaved. However, we are able to construct a smooth open set consisting of solutions with small Higgs field, on which we can investigate the hypersymplectic geometry. Finally, we reinterpret our results in terms of the Riemannian geometry of the moduli space of $G$-connections.