On the motives of moduli of chains and Higgs bundles

TL;DR

This paper introduces a new approach to computing the cohomology of Higgs bundle moduli spaces using localization and motives, providing explicit formulas and supporting conjectures in the field.

Contribution

It presents a novel localization method in the Grothendieck ring, describes motives of moduli stacks of chains, and offers explicit formulas for specific Higgs bundle moduli spaces.

Findings

The n-torsion of the Jacobian acts trivially on the middle cohomology.

Explicit motive formula for rank 4 Higgs bundles with odd degree.

Recursion formulas for motives of stable chain moduli spaces.

Abstract

We take another approach to Hitchin's strategy of computing the cohomology of moduli spaces of Higgs bundles by localization with respect to the circle-action. Our computation is done in the dimensional completion of the Grothendieck ring of varieties and starts by describing the classes of moduli stacks of chains rather than their coarse moduli spaces. As an application we show that the n-torsion of the Jacobian acts trivially on the middle dimensional cohomology of the moduli space of twisted SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for the motive of the moduli space of Higgs bundles of rank 4 and odd degree. This provides new evidence for a conjecture of Hausel and Rodr\'iguez-Villegas. Along the way we find explicit recursion formulas for the motives of several types of moduli spaces of stable chains.