A Universal Family of Deformations for the Uniformising Higgs bundle

TL;DR

This paper constructs a universal deformation family for the uniformising Higgs bundle on hyperbolic Riemann surfaces, providing explicit holomorphic coordinates and extending Kuranishi theory concepts.

Contribution

It introduces a universal analytic deformation family for the uniformising Higgs bundle, enabling explicit holomorphic Darboux coordinates near the Hitchin section.

Findings

Established a unique C*-fixed point on the Hitchin section.

Constructed a universal deformation family with holomorphic Darboux coordinates.

Extended deformation theory using Kuranishi-like constructions.

Abstract

Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of deformations which provides holomorphic Darboux coordinates in a neighbourhood of the section. This is a special case of a more general deformation-theoretic construction in the spirit of Kuranishi theory. As a toy example of the latter we consider the tautological family of centralisers over the Kostant slice.