Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

TL;DR

This paper establishes a correspondence between parabolic G-Higgs bundles on punctured surfaces and representations of the fundamental group into real reductive Lie groups, revealing geometric and algebraic structures of the moduli spaces.

Contribution

It introduces a new correspondence linking parabolic G-Higgs bundles with fundamental group representations for real groups, including boundary cases and special group types.

Findings

Relation between parabolic degree and Tits boundary geometry

Treatment of monodromy logarithm on Weyl alcove boundary

Descriptions of moduli space features for split real and Hermitian groups

Abstract

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.