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

Abstract

We study the mirror symmetry on the singular locus of the Hitchin system at two levels. First, by covering it by (supports of) (BBB)-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup 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 P. Second, when 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.

Fibrados de Higgs e aplicações

“…Em casos mais gerais, de acordo com a literatura física, a simetria do espelho sugere que também deve existir uma correspondência entre (𝐵, 𝐴, 𝐴)-branas em ℳ 𝐺 e (𝐵, 𝐵, 𝐵)branas em ℳL 𝐺 . Essa correspondência em particular foi estudada por Hitchin [35], e mais recentemente em [18] [19]. Uma Ąbra suave da Ąbração de Hitchin é uma (𝐵, 𝐴, 𝐴)-brana e corresponde a feixes skycraper de pontos sobre a Ąbra correspondente em ℳL 𝐺 [42] [31].…”

Linear submanifolds and strata of k-differentials : invariants and applications

“…A brane is a pair (N , F ) of a subvariety N and a sheaf F supported on N with special geometric properties. This initiated plenty of mathematical research to find examples of branes or their supports for G-Hitchin system [BS16]; [Hit16]; [Hit17]; [HS18]; [BS19]; [FJ22]; [HH22]; [FP23]. However, [KW07] also propose a correspondence between branes under mirror symmetry.…”

“…However, [KW07] also propose a correspondence between branes under mirror symmetry. This seems to be less considered in the mathematical literature (see [Hit16]; [FJ22]; [HH22]; [FP23] for exceptions).…”

“…Other than that, we hope to return to this type of visible Lagrangians in a subsequent work and will mainly focus on G = SL(2, C) in the remainder of the paper. The second type of visible Lagrangians appeared in the work of [FGOP21]; [FP23]; [FHHO23]. They are completely contained in the singular locus of the Hitchin map and substantially use the geometry of the singular Hitchin fibers.…”