Fock bundles and Hitchin components

TL;DR

This paper introduces Fock bundles as a new gauge-theoretic tool for studying surface group character varieties, establishing a connection to Hitchin components through solutions of a non-linear PDE.

Contribution

It defines Fock bundles without fixing a complex structure, constructs a canonical flat connection, and links higher complex structures to Hitchin components via explicit solutions.

Findings

Constructed solutions in the Fuchsian locus.

Established a map from higher complex structures to Hitchin components.

Proved ellipticity of the linear operator involved.

Abstract

We introduce the concept of a Fock bundle, a smooth principal bundle over a surface equipped with a special kind of adjoint-valued 1-form, as a new tool for studying character varieties of surface groups. Although similar to Higgs bundles, the crucial difference is that no complex structure is fixed on the underlying surface. Fock bundles are the gauge-theoretic realization of higher complex structures. We construct a canonical connection to a Fock bundle equipped with compatible symmetric pairing and hermitian structure. The space of flat Fock bundles maps to the character variety of the split real form. Determining the hermitian structure such that this connection is flat gives a non-linear PDE similar to Hitchin's equation. We explicitly construct solutions for Fock bundles in the Fuchsian locus. Ellipticity of the relevant linear operator provides a map from a neighborhood of the Fuchsian locus in the space of higher complex structures modulo higher diffeomorphisms to a neighborhood of the Fuchsian locus in the Hitchin component.