Higgs bundle, isomonodromic leaves and minimal surfaces

TL;DR

This paper constructs a gauge-theoretic moduli space of stable G-Higgs bundles on Riemann surfaces, revealing deep relationships between isomonodromic foliation, symplectic structures, and harmonic map energy, leading to new pseudo-Kähler metrics.

Contribution

It introduces a novel gauge-theoretic framework for the joint moduli space of G-Higgs bundles over varying Riemann surfaces, connecting multiple geometric structures.

Findings

Relationship between isomonodromic foliation and symplectic structures

Construction of pseudo-Kähler metrics on character varieties

Recovery of recent higher Teichmüller space results

Abstract

In this paper we give a gauge theoretic construction of the joint moduli space of stable G-Higgs bundles on closed Riemann surfaces, where the Riemann surface structure is allowed to vary in the Teichm\"uller space of the underlying smooth surface. This joint moduli space has many interesting structures that are preserved by the mapping class group of the surface. We describe a surprising relationship between four key objects: the isomonodromic foliation, a canonical hermitian form arising from the Atiyah-Bott-Goldman symplectic structure on the character variety, a canonical holomorphic form which vertically lifts vector fields on Teichm\"uller space, and the energy function for equivariant harmonic maps. One consequence of this work is the construction of pseudo-K\"ahler metrics on many examples of components of character varieties which include rank two higher Teichm\"uller spaces. This recovers some of the recent work on the subject by various authors.