Betti numbers of the moduli space of Higgs bundles over a real curve

TL;DR

This paper derives a formula for the $Z_2$-Betti numbers of the moduli space of stable real Higgs bundles over a real projective curve, expanding understanding of their topological invariants.

Contribution

It introduces a new formula for Betti numbers of real Higgs bundle moduli spaces using motivic measures and existing motivic formulas, bridging real and complex cases.

Findings

Derived a formula for $Z_2$-Betti numbers of real Higgs moduli spaces.

Connected motivic formulas with topological invariants of real algebraic varieties.

Extended motivic measure techniques to real curve Higgs bundle moduli.

Abstract

We produce a formula for the $\mathbb{Z}_2$-Betti numbers of the moduli space $M_r^d$ of stable real Higgs bundles over a real projective curve, with coprime rank $r$ and degree $d$. Our approach relies on the motivic formula for the moduli space due to Mellit, Fedorov-Soibelman-Soibelman, and Schiffmann , and the fact that the virtual $\mathbb{Z}_2$ Poincar\'e polynomial is a motivic measure over $\mathbb{R}$.