Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups

TL;DR

This paper explores mirror symmetry on the singular locus of the Hitchin system by constructing and analyzing branes associated with parabolic subgroups, providing evidence for their duality under mirror symmetry.

Contribution

It introduces a new approach to studying mirror symmetry on the Hitchin system's singular locus using branes related to parabolic subgroups and constructs hyperholomorphic bundles in this context.

Findings

Support of (BBB)-branes corresponds to Higgs bundles with Levi subgroup reduction.

Dual (BAA)-branes are described via Higgs bundles with unipotent radical reduction.

Evidence suggests these branes are dual under mirror symmetry through Fourier-Mukai transforms.

Abstract

We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) $(BBB)$-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup $\mathrm{P}$, whose conjectural dual $(BAA)$-branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of $\mathrm{P}$. Secondly, when $\mathrm{P}$ is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the $(BBB)$-brane, and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an integral functor induced by Fourier-Mukai in the moduli stack of Higgs bundles.