Motivic Integration on the Hitchin Fibration

TL;DR

This paper demonstrates that the moduli spaces of twisted SL_n and PGL_n-Higgs bundles on a smooth projective curve share the same stringy class in the Grothendieck ring, confirming a conjecture related to their Hodge numbers.

Contribution

It introduces a motivic integration approach with Chow motives to compare Hitchin fibers, extending previous p-adic methods and confirming conjectures about their classes.

Findings

Moduli spaces of twisted SL_n and PGL_n-Higgs bundles have identical stringy classes.

The motivic integration method confirms the conjectured equality of Hodge numbers.

The approach uses Néron models to evaluate integrals on Hitchin fibers.

Abstract

We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently proven by Groechenig, Ziegler and the second author. To adapt their argument, which relies on p-adic integration, we use a version of motivic integration with values in rational Chow motives and the geometry of N\'eron models to evaluate such integrals on Hitchin fibers.