Quantum geometric Langlands at irrational level
From OpenAI Aftermath
Quantum geometric Langlands at irrational level concerns the conjectured equivalence between categories of twisted D-modules on the stack of G-bundles on a curve and on the stack of bundles for the Langlands dual group, at dual quantum parameters. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 4 October 2026, claims this equivalence for every irrational level. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The quantum form of geometric Langlands compares twisted D-module categories at inverse shifted levels. Kapustin and Witten's gauge-theoretic interpretation explains the dual root datum and the reciprocal parameter; an early formulation, attributed to Drinfeld, is Stoyanovsky's inverse-parameter conjecture; and the formulation through local categories, localization and Whittaker coefficients is developed in Gaitsgory's quantum Langlands programme.
- Local and classical inputs include Feigin–Frenkel duality of W-algebras, the work of Arakawa and Frenkel, Raskin's categorical Drinfeld–Sokolov reduction, the fundamental local equivalences of Campbell, Dhillon and Raskin, and Gaitsgory's local–global Whittaker comparison. On the global side the manuscript uses the proof of the ordinary geometric Langlands conjecture by Gaitsgory and coauthors.
- Bogdanova constructed a quantum Langlands functor in the Betti setting and specified its central compatibility.
The original result
- Claim. Theorem 1.1 of Global quantum geometric Langlands at irrational level states that for every connected simple complex algebraic group G, every smooth projective connected complex curve X and every level c in C minus Q, there is an equivalence of presentable DG categories between the twisted D-modules on BunG(X) at level c and those on BunG∨(X) at level −1/(rc), where r is the lacing number. The manuscript states that the equivalence keeps the given global forms and all connected components, is characterized by a localization–Whittaker comparison, and intertwines central torsor actions with transgression local systems; it presents this as the unramified de Rham case, in the irrational range, of the conjecture stated as Conjecture 1 in Bogdanova's paper on the Betti quantum Langlands functor.
- Method, as described: localization is a quotient functor, so the two sides are first compared on localized objects through local pairings and a global period identity. The two main inputs are a comparison between global Whittaker coefficients and a Poincaré functor, proved at irrational exponent by a deformation argument with weights and Kummer local systems, and local pairings over families of marked points, built from Drinfeld–Sokolov reduction and inverse-level W-algebra duality and extended across collision diagonals.
- Source (openai/math repository, commit
7ef5c0c): Global quantum geometric Langlands at irrational level. - Formalization. The collection lists no Lean formalization for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 28 October 2026; AI Proved What? row 28 October 2026) |
|---|---|---|---|---|
| 2026-10-10 | F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, OSVerify, Stanford University: referee report on Global quantum geometric Langlands at irrational level | Review, referee report | Reviews the manuscript version of 4 October 2026 and reports two major issues and one minor issue. First, the absolute pairing in Proposition 4.3 is asserted to be continuous over parameter spaces that need not be quasi-compact; a counterexample at exponent zero, which the report says is not a counterexample at irrational level, shows that it fails to preserve direct sums, and the proposed repair asks for a separate continuous evaluation that the report does not construct in general. Second, in the proof of Proposition 6.1 constant fibre cohomology is taken to give a global scalar identification, which a counterexample for SL2 refutes; the report proposes a descent argument that still gives the conservativity needed later. The minor issue is three cross-references to the wrong equation. | Web page at osverify.github.io, first posted 10 October 2026, last changed 12 October 2026. The page states that its findings and severity labels come from OSVerify's internal review and do not necessarily establish that the main theorem is false; the site's home page states that every reported issue was verified by human experts. |
| 2026-10-13 | AI Proved What?, episode 069 Langlands, quantized?, remade version of 26 October 2026 | Reading (exposition) | Video explainer of the claim that for every connected simple complex group, every smooth projective complex curve and every level c that is not rational, including non-real levels, the twisted D-module category on Bun of G at level c is equivalent to that on Bun of the dual group at level −1/rc, keeping the actual global forms and all connected components. The episode states that rational levels, reductive groups that are not simple, such as GL n, and ramified versions are not covered. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 19 October 2026: posted as a preprint: OpenAI, dated October 4, 2026, released October 6; under expert review: not yet; peer reviewed: not yet, a public review by OSVerify lists two major issues and one minor one, each with a proposed repair; checked in Lean: no, OpenAI published no Lean proof for this result. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
