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

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.