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arxiv: 2606.29585 · v1 · pith:FDOWGDS6new · submitted 2026-06-28 · 🧮 math.RT · math.AG

Quantum Betti geometric Langlands functor

Ekaterina Bogdanova This is my paper

Pith reviewed 2026-06-30 01:40 UTC · model grok-4.3

classification 🧮 math.RT math.AG
keywords quantum geometric LanglandsBetti settingWhittaker coefficients2-Fourier-Mukai equivalencegerbescenter of Gfundamental group of dual groupsheaves of categories
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The pith

The quantum geometric Langlands functor is constructed in the Betti setting via Whittaker coefficients and shown compatible with the 2-Fourier-Mukai equivalence on gerbe 2-stacks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the quantum geometric Langlands functor in the Betti setting by using Whittaker coefficients as the defining mechanism. It then verifies that this functor respects the 2-Fourier-Mukai equivalence between sheaves of categories over two 2-stacks that classify gerbes on a curve X, one with respect to the center of G and the other with respect to the algebraic fundamental group of the dual group. A sympathetic reader would care because the construction supplies an explicit bridge between quantum versions of automorphic and spectral sides in the Betti framework, where previous approaches lacked such a direct functor. If the claim holds, it would give a concrete way to move data between the two sides while preserving the duality encoded in the gerbe stacks.

Core claim

We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks Ge_{Z_G} and Ge_{π_1(Ĝ)}, which classify gerbes on X with respect to the center Z_G of G and algebraic fundamental group π_1(Ĝ) of Ĝ.

What carries the argument

Whittaker coefficients, which define the quantum geometric Langlands functor and are used to verify its compatibility with the 2-Fourier-Mukai equivalence between the sheaves of categories on the two gerbe 2-stacks.

If this is right

  • The functor supplies an explicit quantum map from one side of the geometric Langlands correspondence to the other in the Betti context.
  • Compatibility with the equivalence ensures the map respects the duality between center and fundamental group gerbes.
  • Whittaker coefficients become the practical tool for computing the functor on objects.
  • The construction extends the classical geometric Langlands functor to its quantum version while preserving the stacky structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Whittaker-based approach might apply in de Rham or other geometric settings if the corresponding 2-stacks and equivalences can be defined there.
  • Checking the functor on unipotent representations or on specific automorphic forms would provide concrete test cases.
  • The result suggests that gerbe data on the two sides can be interchanged systematically in quantum settings.

Load-bearing premise

The 2-stacks Ge_{Z_G} and Ge_{π_1(Ĝ)} classify gerbes on X with respect to Z_G and π_1(Ĝ), and the 2-Fourier-Mukai equivalence between the corresponding sheaves of categories is available in the Betti setting.

What would settle it

An explicit computation for a low-rank group such as SL(2) on a specific curve X that produces a mismatch between the Whittaker-defined functor and the image under the 2-Fourier-Mukai equivalence.

read the original abstract

We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks $\operatorname{Ge}_{Z_G}$ and $\operatorname{Ge}_{\pi_1(\check{G})}$, which classify gerbes on $X$ with respect to the center $Z_G$ of $G$ and algebraic fundamental group $\pi_1(\check{G})$ of $\check{G}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. It further asserts that this functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over the 2-stacks Ge_{Z_G} and Ge_{π_1(Ĝ)}, which classify gerbes on X with respect to the center Z_G of G and the algebraic fundamental group π_1(Ĝ) of Ĝ.

Significance. If the stated construction and compatibility were fully detailed and verified, the result would contribute to the quantum geometric Langlands program in the Betti setting by linking Whittaker coefficients to functoriality across the indicated 2-stacks. However, the absence of any derivations, definitions, or proof steps in the provided manuscript prevents evaluation of whether the claims hold or advance the field beyond existing background facts on the 2-stacks and equivalence.

major comments (1)
  1. The manuscript consists solely of the abstract with no sections, equations, definitions of Whittaker coefficients in this context, or proof steps. This makes it impossible to verify the construction or the compatibility claim, as no technical content is available to assess soundness or internal consistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We acknowledge that the submitted version contained only the abstract and will expand the manuscript substantially in revision to include the full construction, definitions, and proofs.

read point-by-point responses

  1. Referee: The manuscript consists solely of the abstract with no sections, equations, definitions of Whittaker coefficients in this context, or proof steps. This makes it impossible to verify the construction or the compatibility claim, as no technical content is available to assess soundness or internal consistency.

    Authors: The referee is correct that the current submission is limited to the abstract. The revised manuscript will contain dedicated sections with the definition of the quantum geometric Langlands functor via Whittaker coefficients in the Betti setting, the relevant 2-stacks Ge_{Z_G} and Ge_{π_1(Ĝ)}, the 2-Fourier-Mukai equivalence, and the detailed proof of compatibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents a construction of the quantum geometric Langlands functor via Whittaker coefficients together with a compatibility statement that uses the 2-Fourier-Mukai equivalence on the indicated 2-stacks as a background fact. No equations, fitted parameters, self-definitions, or load-bearing self-citations appear in the text that would reduce any claimed result to its own inputs by construction. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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discussion (0)

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Reference graph

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