Stacky dualities for the moduli of Higgs bundles

TL;DR

This paper establishes a duality between moduli stacks of Higgs bundles for Langlands dual groups, revealing new symmetries and connections to physical theories through hyperkähler geometry and B-fields.

Contribution

It identifies the shifted Cartier dual of Higgs bundle moduli stacks with dual stacks, linking Hitchin systems via hyperkähler rotation and proposing new self-dual stacks related to Coulomb branches.

Findings

Identification of shifted Cartier dual with Langlands dual stacks

Connection of Hitchin systems through hyperkähler rotation

Existence of self-dual stacks conjectured as Coulomb branches

Abstract

The central result of this paper is an identification of the shifted Cartier dual of the moduli stack $\mathcal{M}_{\mathfrak{g}}(C)$ of $\widetilde{G}$-Higgs bundles on $C$ of arbitrary degree (modulo shifts by $Z(\widetilde{G})$) with a quotient of the Langlands dual stack $\mathcal{M}_{^L\mathfrak{g}}(C)$. Via hyperk\"ahler rotation, this may equivalently be viewed as the identification of an SYZ fibration relating Hitchin systems for arbitrary Langlands dual semisimple groups, coupled to nontrivial finite $B$-fields. As a corollary certain self-dual stacks $\frac{\mathcal{M}_{\mathfrak{g}}(C)}{\Gamma}$ are observed to exist, which I conjecture to be the Coulomb branches for the 3d reduction of the 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$.