Homotopy Type of Moduli Spaces of G-Higgs Bundles and Reducibility of the Nilpotent Cone

TL;DR

This paper investigates the topology of moduli spaces of G-Higgs bundles over higher genus surfaces, revealing obstructions to deformation retraction and demonstrating the reducibility of the nilpotent cone for complex G.

Contribution

It introduces new obstructions to deformation retraction in higher genus cases and proves the nilpotent cone's reducibility for complex reductive groups.

Findings

Obstructions to deformation retraction for higher genus surfaces.

Reducibility of the nilpotent cone in the moduli space.

Contrast with genus one case where retraction is possible.

Abstract

Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the complexification of its maximal compact subgroup $H\subset G$. We consider classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus $g\geq2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles over $X$, in contrast with the situation when $g=1$, and to show reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$ complex.