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Global quantum geometric Langlands at irrational level
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Abstract
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level $c\in\mathbb C\setminus\mathbb Q$, including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on $\mathop{\mathrm{Bun}}\nolimits _G(X)$ and $\mathop{\mathrm{Bun}}\nolimits _{G^\vee}(X)$ at the dual shifted levels c and $-1/(rc)$, where r is the lacing number. The equivalence uses the given global forms and includes all connected components.
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Comments and discussion
- OSVerify, Global quantum geometric Langlands at irrational level (Paper Referee report on the October 4, 2026 version; accessed 2026-10-21). The report, described by OSVerify as further verified by human experts, flags failure of continuity for Proposition 4.3's absolute pairing over non-quasi-compact parameters (with a counterexample at exponent zero), an unsupported global scalar identification in Proposition 6.1, and incorrect cross-references, while proposing repairs and explicitly not claiming to disprove the main theorem. [AI-generated per source]
Related prior literature mentioned by others
- Justin Campbell, Gurbir Dhillon and Sam Raskin, Fundamental local equivalences in quantum geometric Langlands (arXiv:1907.03204; Compositio Mathematica 157, 2021). The AI Proved What? exposition identifies these local equivalences as prior inputs to the manuscript's claimed global equivalence.
- Ekaterina Bogdanova, Quantum Betti geometric Langlands functor (arXiv:2606.29585, 2026-06-28). The AI Proved What? exposition highlights this earlier construction of a Betti quantum Langlands functor via Whittaker coefficients, with central compatibility, as context for the manuscript's claimed de Rham equivalence.
Possibly relevant literature (found by the agent, not checked)
- Yifei Zhao, Quantum parameters of the geometric Langlands theory (Selecta Mathematica 29, article 66, 2023-08-22). Example 2.15 gives the shifted parameter transformation from c to −1/(rc), providing background for the normalization and dual levels in the manuscript's Theorem 1.1.
- D. Gaitsgory and S. Lysenko, Parameters and duality for the metaplectic geometric Langlands theory (arXiv:1608.00284, 2016, rev. 2022). Describes quantum parameters as factorization gerbes on the affine Grassmannian and develops the corresponding duality; relevant to the paper's level normalization and its exchange of central characters and components between G and G^∨.
- Edward Frenkel and Davide Gaiotto, Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks (arXiv:1805.00203, 2018; Commun. Number Theory Phys. 14, 2020). Builds twisted D-modules on Bun_G from conformal blocks of junction vertex algebras for every simple G and treats spin and gerbe structures, which bears on the paper's claim that the equivalence keeps the given global forms and intertwines central torsor actions.
- Thomas Creutzig and Davide Gaiotto, Vertex Algebras for S-duality (arXiv:1708.00875, 2017). Proposes kernel vertex algebras containing two affine subalgebras at duality-related levels, whose chiral homology should reproduce quantum geometric Langlands; compare with the paper's local pairings of Kac–Moody modules and Whittaker objects through W-algebra duality.
- Mina Aganagic, Edward Frenkel and Andrei Okounkov, Two-parameter Langlands Correspondence (arXiv:2510.20948, 2025). A survey of the q-deformed quantum Langlands correspondence that proposes a non-simply-laced extension; relevant to the lacing-number factor r in the paper's dual level -1/(rc).
Digests and expositions
- AI Proved What?, Langlands, quantized? Quantum geometric Langlands at irrational level (Episode 069, October 2026; status checked 2026-10-12). This video and full transcript outline the claimed equivalence, the uses of irrationality, the local-to-global construction, and OSVerify's reported issues. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 069
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026,
author = {{OpenAI}},
title = {{Global quantum geometric Langlands at irrational level}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf}{OAI:Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026}},
year = {2026}
}Family 069: Global quantum geometric Langlands at irrational level
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level $c\in\mathbb C\setminus\mathbb Q$. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.
This is the only paper in this family.