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The central result of this paper is an identification of the shifted Cartierdual 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 mayequivalently be viewed as the identification of an SYZ fibration relatingHitchin systems for arbitrary Langlands dual semisimple groups, coupled tonontrivial finite $B$-fields. As a corollary certain self-dual stacks$\frac{\mathcal{M}_{\mathfrak{g}}(C)}{\Gamma}$ are observed to exist, which Iconjecture to be the Coulomb branches for the 3d reduction of the 4d$\mathcal{N}=2$ theories of class $\mathcal{S}$.

Stacky dualities for the moduli of Higgs bundles