Result 069, Algebraic and complex geometry

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∈C∖Qc\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.

Proof

The bigger picture

Why it matters

Two different symmetry groups can organize the same geometry of differential equations. The manuscript claims a precise version of this principle for Langlands-dual groups, providing a way to translate mathematical questions between their geometric settings.

What changes?

The manuscript reports an unramified de Rham equivalence for every connected simple complex algebraic group G on every smooth projective connected complex curve X. It matches full derived categories of twisted D-modules, geometric systems of differential equations on spaces parametrizing G-bundles and Langlands-dual-group bundles. Every shifted complex level c outside the rational numbers is allowed, including non-real levels; the dual level is -1/(r c), with r the group's lacing number. It retains the given global forms and all connected components.

What does that help mathematicians do?

An equivalence of full derived categories preserves not just individual objects but also maps and higher relationships between them. Researchers can therefore translate questions about these differential-equation objects to the dual group's setting without discarding components of the bundle space. Keeping the specified global forms matters because the claimed correspondence respects the actual groups chosen, rather than only a restricted version of their geometry.

Are there practical applications?

The immediate value is foundational: the claimed equivalence provides a common framework for studying bundle geometry and differential equations for dual groups. Its scope is unramified, with no added ramification data at points of the curve. The supplied material establishes no computational method or practical deployment, and the claim does not cover rational shifted levels.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Global quantum geometric Langlands at irrational level

October 4, 2026 119 pages

We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level c∈C∖Qc\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 BunG(X)\mathop{\mathrm{Bun}}\nolimits _G(X) and BunG∨(X)\mathop{\mathrm{Bun}}\nolimits _{G^\vee}(X) at the dual shifted levels c and −1/(rc)-1/(rc), where r is the lacing number. The equivalence uses the given global forms and includes all connected components.

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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.