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

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 $\mathbb{Z}_2$ Poincaré polynomial is a motivic measure over $\mathbb{R}$.

💡 Research Summary

This paper establishes an explicit formula for the Z₂-Betti numbers of the moduli space M_r^d of stable real Higgs bundles over a real projective curve. The Higgs bundles are of coprime rank r and degree d on a smooth, projective, geometrically connected real curve Σ of genus g. The real locus Σ(R) is assumed to be a union of b+1 circles.

The core strategy leverages powerful motivic techniques from complex geometry and adapts them to the real setting. The starting point is the “motivic formula” for the moduli space over a field of characteristic zero, developed by Schiffmann, Mellit, and Fedorov-Soibelman-Soibelman. This formula expresses the class