Betti geometric Langlands conjecture

Let $G$ be a complex reductive group, let $X$ be a smooth complex projective curve, let $G^{\vee}$ be its Langlands dual group, and let $\mathcal Loc_{G^{\vee}}(X)$ be the Betti moduli stack of $G^{\vee}$ -local systems. Let $\mathcal Shv_{\mathcal N}(Bun_G(X))$ be the dg category of nilpotent sheaves, and let $\mathcal QC^!_{\mathcal N}(\mathcal Loc_{G^{\vee}}(X))$ denote ind-coherent sheaves with nilpotent singular support. Betti geometric Langlands conjecture. There is an equivalence of dg categories

\mathcal Shv_{\mathcal N}(Bun_G(X))\simeq \mathcal QC^!_{\mathcal N}(\mathcal Loc_{G^{\vee}}(X))

compatible with actions of Hecke functors. It is the Betti analogue of the de Rham conjecture, replacing the de Rham moduli space by the topological character stack and the automorphic category by sheaves with nilpotent singular support.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Betti geometric Langlands conjecture
Let $G$ be a reductive group, let $\Bun_G$ denote the moduli stack of $G$ -bundles, and let $\LS^{\on{Betti}}_\cG$ be the Betti stack of local systems. Let $\IndCoh_\Nilp(\LS^{\on{Betti}}_\cG)$ denote the category of ind-coherent sheaves with nilpotent singular support, and let
$\BL^{\on{Betti}}_G:\Shv^{\on{Betti}}_{\frac{1}{2},\Nilp}(\Bun_G)\to \IndCoh_\Nilp(\LS^{\on{Betti}}_\cG)$
be the continuous geometric Langlands functor characterized in the Betti context. The Betti geometric Langlands conjecture. The functor $\BL^{\on{Betti}}_G$ is an equivalence.

This is the Betti formulation of geometric Langlands, originally formulated by D. Ben-Zvi and D. Nadler. The functor is constructed and shown to have the stated characterization, while the equivalence remains conjectural.
source: Dennis Gaitsgory and Sam Raskin, “Proof of the geometric Langlands conjecture I: construction of the functor”, arXiv:2405.03599 (2025).

Sources & referencesView supporting material

Primary source

David Ben-Zvi and David Nadler, “Betti Geometric Langlands”, arXiv:1606.08523 (2016).

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Betti geometric Langlands conjecture

Equivalent formulations 1

The Betti geometric Langlands conjecture

Primary source

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