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The global quantum geometric Langlands conjecture at irrational level

For a connected simple complex group GG with Langlands dual G∨G^\vee, lacing number rr and a smooth projective curve XX, let Dc(BunG(X))D_c(\mathrm{Bun}_G(X)) be the DG category of D-modules on the moduli stack of GG-bundles twisted by the power of the determinant line corresponding to shifted level cc. The quantum geometric Langlands conjecture, in Stoyanovsky's inverse-parameter form attributed to Drinfeld and developed in Gaitsgory's quantum Langlands program, predicts an equivalence Dc(BunG(X))≃D−1/(rc)(BunG∨(X))D_c(\mathrm{Bun}_G(X))\simeq D_{-1/(rc)}(\mathrm{Bun}_{G^\vee}(X)) compatible with localization and Whittaker coefficients, deforming ordinary geometric Langlands. Does this equivalence hold for every level cc, in particular for irrational cc?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Geometric representation theory; geometric Langlands
Posed by
A. V. Stoyanovsky (attributing the idea to Drinfeld); developed by Dennis Gaitsgory
Year posed
2006
Years open
20y
Solved
2026-10-04
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every connected simple complex algebraic group GG, every smooth projective connected complex curve XX and every c∈C∖Qc\in\mathbb C\setminus\mathbb Q (including non-real cc) there is an equivalence Dc(BunG(X))≃D−1/(rc)(BunG∨(X))D_c(\mathrm{Bun}_G(X))\simeq D_{-1/(rc)}(\mathrm{Bun}_{G^\vee}(X)) of presentable DG categories, characterized by its localization-Whittaker comparison and intertwining central torsor actions with transgression local systems. It does not treat rational levels, ramification, the Betti setting, or non-simple reductive groups.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated October 4, 2026. The paper uses the proof of ordinary geometric Langlands and the local equivalences of Campbell-Dhillon-Raskin as inputs.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture in the cited normalization; it gives the unramified de Rham equivalence for every connected simple GG, every smooth projective connected curve and every c∈C∖Qc\in\mathbb C\setminus\mathbb Q, with given global forms and all connected components, characterized by a localization-Whittaker comparison. Scope limits stated by the paper: irrational levels only (rational levels, including the critical and integral ones, are excluded), unramified only, de Rham setting. The argument depends on the proof of ordinary geometric Langlands (Gaitsgory, Raskin and coauthors) and on other large published inputs. No Lean formalization exists for this family. The proof was not refereed.

Sources

Changelog1 change

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