Unramified covers and branes on the Hitchin system

TL;DR

This paper explores the geometry of Higgs bundles on curves via unramified covers, constructing dual branes in the Hitchin system and analyzing their behavior under mirror symmetry, especially over singular fibers.

Contribution

It introduces new BBB- and BAA-branes associated with unramified covers on the Hitchin system and demonstrates their duality through explicit Fourier--Mukai transforms, linking to mirror symmetry.

Findings

Branes are constructed over the singular locus of the Hitchin fibration.

Explicit Fourier--Mukai transform establishes duality of these branes.

The work relates to the mirror symmetry of fixed loci under tensorization actions.

Abstract

We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier--Mukai transform, where we recall that the structure group $\mathrm{GL}(n,\mathbb{C})$ is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. As a particular case, our construction describes the behaviour under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order $n$. These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.