Global quantum geometric Langlands at irrational level

1 · The essentials

The manuscript describes a precise correspondence between two mathematical descriptions built on the same complex curve. One starts with a group that encodes symmetry; the other uses its designated dual group. Each description also depends on a numerical setting called a level. The correspondence is shown in the manuscript for every allowed group and curve, whenever the shifted level is not a rational number, including when it is non-real. Changing to the dual group requires changing this setting by a specified rule. The claim covers the full collections of objects under study, without discarding any of their connected pieces. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two catalogues describing the same collection in different languages, each with its own indexing setting. The claimed correspondence translates between the entire catalogues, but only when their settings are linked by the required rule.

Why it matters

The significance of the claim is its scope: it identifies the full derived categories in the stated setting, rather than only selected parts. It also retains the given global forms of the groups and all connected components. The permitted levels include non-real values, not just irrational real values.

2 · Going further

The precise claim is an unramified de Rham quantum geometric Langlands equivalence. Let $G$ be any connected simple complex algebraic group, let $G^{\lor}$ be its Langlands dual, and let $X$ be any smooth projective connected complex curve. For every shifted level $c \in C ∖ Q$ , the manuscript claims an equivalence between the full derived categories of twisted $D$ -modules on $Bun_{G} (X)$ and $Bun_{G^{\lor}} (X)$ .

The shifted levels on the two sides are $c$ and $- 1/ (r c)$ , where $r$ is the lacing number. Thus the claim pairs a change of group with a specific change of level. The source explicitly says that the equivalence uses the given global forms and includes all connected components; it is not presented as a statement restricted to selected components.

The supplied abstract states the result but does not describe a construction of the equivalence or the main idea of its argument. Only the main manuscript is supplied, so no relationship with companion manuscripts can be described from this source. The result is shown in the manuscript according to the supplied account, but it is not formalized in Lean and has not been formally verified.

Useful notions

Complex algebraic groups and their global forms
Smooth projective complex curves
Moduli of bundles
Derived categories and twisted D-modules
Langlands dual groups and shifted levels

What was known before / what is new

Before

The source places the result within unramified de Rham quantum geometric Langlands. It supplies no account of earlier results, previously established special cases, or the historical development of the problem.

New

The manuscript claims the equivalence for every connected simple complex algebraic group and every smooth projective connected complex curve at every shifted level in $C ∖ Q$ . It explicitly includes non-real levels. Its stated scope encompasses the full derived categories, the given global forms, and all connected components.

Limits and assumptions

The claim is restricted to the unramified de Rham setting, connected simple complex algebraic groups, and smooth projective connected complex curves. The shifted level must lie in $C ∖ Q$ , and the dual shifted level is prescribed as $- 1/ (r c)$ . Rational levels and settings outside these hypotheses are not covered by the supplied statement; the source does not say what remains open there. The abstract supplies no argument details. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Langlands dual group: The group denoted $G^{\lor}$ that is paired with $G$ in the stated equivalence.
Shifted level: The parameter specifying the levels in the statement, with $c$ on the $G$ side and $- 1/ (r c)$ on the dual side.
Lacing number: The quantity denoted $r$ in the formula for the dual shifted level.
Twisted D-module categories: The categories on the two bundle spaces whose full derived versions are identified by the claimed equivalence.
Connected components: The connected pieces of the spaces under study, all of which are included in the stated result.

Keywords : Quantum geometric Langlands · Twisted D-modules · Irrational level

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 $c \in C ∖ Q$ . It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Global quantum geometric Langlands at irrational level

04/10/2026

To be verified

We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level $c \in C ∖ Q$ , including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on $Bun_{G} (X)$ and $Bun_{G^{\lor}} (X)$ at the dual shifted levels c and $- 1/ (r c)$ , where r is the lacing number. The equivalence uses the given global forms and includes all connected components.

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