Mirror symmetry for parabolic Higgs bundles via $p$-adic integration

TL;DR

This paper proves the topological mirror symmetry conjecture for moduli spaces of parabolic Higgs bundles using $p$-adic integration, extending previous results to the parabolic case and analyzing $E$-polynomials.

Contribution

It establishes the mirror symmetry for parabolic Higgs bundles and shows the $E$-polynomial independence from degree, advancing the understanding of these moduli spaces.

Findings

Proof of the topological mirror symmetry conjecture for parabolic Higgs bundles.

Demonstration that the $E$-polynomial is independent of the degree.

Extension of previous non-parabolic results to the parabolic setting.

Abstract

Applying the technique of $p$-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups $\text{SL}_n$ and $\text{PGL}_n$, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the $E$-polynomial of the smooth moduli space of parabolic $\text{GL}_n$-Higgs bundles is independent of the degree of the underlying vector bundles.