Unramified covers and branes on the Hitchin system

Abstract

We study the locus of the moduli space of GL(n,C)-Higgs bundles on a curve givenby those Higgs bundles obtained by pushforward under a connected unramified cover. We equipthese loci with a hyperholomorphic bundle so that they can beviewed as BBB-branes, and weintroduce corresponding BAA-branes which can be describedvia Hecke modifications. Wethen show how these branes are naturally dual via explicit Fourier–Mukai transform (recall that GL(n,C) is Langlands self dual). It is noteworthy that these branes lie over the singularlocus of the Hitchin fibration.As a particular case, our construction describes the behavior under mirror symmetry of thefixed loci for the action of tensorization by a line bundle of ordern. These loci play a key rolein the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.