Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

TL;DR

This paper investigates the topology of real loci in moduli spaces of Higgs bundles on Riemann surfaces with anti-holomorphic involutions, revealing that the number of connected components matches that of real Picard groups under certain conditions.

Contribution

It introduces a novel analysis of the real structure on Higgs bundle moduli spaces using real and quaternionic Hodge bundles, establishing a correspondence in the number of connected components.

Findings

Number of connected components of real Higgs moduli space equals that of real Picard group when gcd(r,d)=1.

Utilizes real and quaternionic systems of Hodge bundles to study real loci.

Provides topological insights into the structure of moduli spaces with anti-holomorphic involutions.

Abstract

Let $X$ be a compact Riemann surface $X$ of genus $\geqslant 2$ and let $\sigma:X \to X$ be an anti-holomorphic involution. Using real and quaternionic systems of Hodge bundles, we study the topology of the real locus $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ of the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ on $X$, for the induced real structure $(E,\phi) \to (\sigma^*(\overline{E}),\sigma^*(\overline{\phi}))$. We show in particular that, when $\mathrm{gcd}(r,d)=1$, the number of connected components of $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ coincides with that of $\mathbb{R} \mathrm{Pic}_d(X)$, which is well-known.