Unramified covers and branes on the Hitchin system

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.

References in corpus (19)

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Cited by in corpus (8)

• Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups
• Endoscopic decompositions and the Hausel-Thaddeus conjecture
• The Kapustin-Witten equations and nonabelian Hodge theory
• Narasimhan--Ramanan branes and wobbly Higgs bundles
• Equality of the wobbly and shaky loci
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• Fixed point branes, singular loci and mirror symmetry
• Mirror of Orbifold Singularities in the Hitchin Fibration: the case $({\text{SL}}_n,{\text{PGL}}_n)$