Stacky dualities for the moduli of Higgs bundles

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 $\tilde{G}$-Higgs bundles on $C$ of arbitrary degree (modulo shifts by $Z(\tilde{G})$) with a quotient of the Langlands dual stack ${\mathcal{M}}_{{}^L\mathfrak{g}}(C)$. Via hyperkähler 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 $\mathcal{S}$.

43 pages; added a section on background for the physical conjecture to the introduction, fixed typos, clarified and expanded certain proofs (results unchanged). To appear in Adv. Math