Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

TL;DR

This paper proves the Topological Mirror Symmetry Conjecture for certain Higgs bundle moduli spaces using p-adic integration and establishes independence of Higgs bundle counts from degree over finite fields.

Contribution

It establishes the Topological Mirror Symmetry Conjecture for SL_n and PGL_n Higgs moduli spaces and proves degree independence of Higgs bundle counts over finite fields.

Findings

Proved equality of stringy Hodge numbers for dual orbifolds.

Used p-adic integration and Tate duality in the proof.

Confirmed degree independence of Higgs bundle counts for coprime degree and rank.

Abstract

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of degree $d$ over a fixed curve defined over a finite field, is independent of $d$. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.