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Math-2595 · mathematics · Algebraic geometry

Unramified de Rham quantum Langlands at irrational level

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every smooth connected projective complex curve X and connected simple complex group G with specified full root datum, lacing number r and Langlands dual $G\vee $, fix dual pinnings and a theta characteristic. Define $D_c(Bun_G)$ as the presentable derived category of all D-modules twisted by the adjoint determinant line to exponent $(c-h\vee )/(2h\vee )$. For each $c\in C$ outside Q, is there a C-linear DG equivalence $D_c(Bun_G(X))\simeq D_(-1/(rc))(Bun_{G\vee }(X))$, using all connected components and compatible with localization–Whittaker comparison, moving-point collisions and the corresponding central-torsor/transgression actions?

Short English statement

For every smooth connected projective complex curve X and connected simple complex group G with specified full root datum, lacing number r and Langlands dual $G\vee $, fix dual pinnings and a theta characteristic. Define $D_c(Bun_G)$ as the presentable derived category of all D-modules twisted by the adjoint determinant line to exponent $(c-h\vee )/(2h\vee )$. For each $c\in C$ outside Q, is there a C-linear DG equivalence $D_c(Bun_G(X))\simeq D_(-1/(rc))(Bun_{G\vee }(X))$, using all connected components and compatible with localization–Whittaker comparison, moving-point collisions and the corresponding central-torsor/transgression actions?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2595-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2595

Manuscript 1 (family 069, 2026-10-04): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Unramified de Rham quantum Langlands at irrational level

OpenAI's claim

At every complex irrational level, the full unramified twisted $D$-module categories for a connected simple group and its dual are claimed to be equivalent at levels $c$ and $-1/(rc)$, including all global-form components. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be a smooth connected projective complex curve and $G$ a connected simple complex algebraic group, with Langlands dual $G^\vee$ of the full dual root datum. On $\operatorname{Bun}_G(X)$ let $L_G|_E=\det R\Gamma(X,\mathfrak g_E)$, and define $D_c=D(L_G^{(c-h^\vee)/(2h^\vee)})$, the cocomplete derived category of all twisted $D$-modules. Fix dual pinnings and a theta characteristic, and let $r\in\{1,2,3\}$ be the lacing number. OpenAI claims for every $c\in\mathbb C\setminus\mathbb Q$ an equivalence \[ D_c(\operatorname{Bun}_G(X))\simeq D_{-1/(rc)}(\operatorname{Bun}_{G^\vee}(X)) \] of presentable complex-linear DG categories. The assertion retains the actual global group forms and all connected components, and includes compatibility with localization--Whittaker comparison under moving and colliding marked points, and with central torsor actions and their transgression local systems. The complex line power specifies differential-operator twisting, not an assumed algebraic root.

Outline of the argument

Manuscript 1 (family 069, 2026-10-04): Local Whittaker comparisons and the classical inputs define the candidate kernel. A weight argument controls its filtered deformation globally, and coherent extension of local pairings across marked-point collisions establishes the localization comparison and compatibility with central torsor actions.

Formal verification

Family 069 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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