nLab categorical geometric Langlands conjecture

Idea

The categorical geometric Langlands conjecture is a higher category theoretic version of the geometric Langlands conjecture. It is formulated as an equivalence of stable $\infty$-categories between:

1. D-modules on the derived stack of G-bundles on a curve $X$,

2. ind-coherent sheaves on the derived stack of $G^\vee$-equivariant local systems on $X$,

where

• $X$ is a smooth complete algebraic curve over a field of characteristic zero,

• $G$ is a reductive group,

• $G^\vee$ denotes its Langlands dual.

This was proven by Dennis Gaitsgory et al. in 2025, cf. the references below.

References

General

• Gurbir Dhillon, An informal introduction to categorical representation theory and the local geometric Langlands program (arXiv:2205.14578)

• Dennis Gaitsgory, Notes on geometric Langlands, web.

• Dennis Gaitsgory, Nick Rozenblyum, A study in derived algebraic geometry.

• Dennis Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.

• A precise statement of the categorical version of geometric Langlands conjecture (2011) [MO:q56571]

Proof

• Dennis Gaitsgory: Proof of the geometric Langlands conjecture (2025) [webpage]

• Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599, pdf]

• Dima Arinkin, D. Beraldo, Justin Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE [arXiv:2405.03648, pdf]

• Justin Campbell, Lin Chen, Joakim Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture III: compatibility with parabolic induction [arXiv:2409.07051, pdf]

• Dima Arinkin, D. Beraldo, L. Chen, J. Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture IV: ambidexterity [arXiv:2409.08670, pdf]

• Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture V: the multiplicity one theorem [arXiv:2409.09856, pdf]

Review:

• Dennis Gaitsgory: On the Proof of the Geometric Langlands Correspondance, IHES News (5 August 2024) [www.ihes.fr/en/gaitsgory-glc]

On whether the proposed interpretation in string theory was of help in the proof:

• Interview with MPIM Director Dennis Gaitsgory (Jan 2026) [video]

6:40-: “It may not necessarily be the case that I understood what [Witten] told me, but somehow, as a result of this conversation, something clicked and we understood the basic structure of how [geometric Langlands] is supposed to work. That was purely inspirational.”