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Quantum Analytic Langlands Correspondence
by Davide Gaiotto, Jörg Teschner
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Davide Gaiotto |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2402.00494v1 (pdf) |
| Date accepted: | April 9, 2025 |
| Date submitted: | Oct. 31, 2024, 11:58 p.m. |
| Submitted by: | Gaiotto, Davide |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The analytic Langlands correspondence describes the solution to the spectral problem for the quantised Hitchin Hamiltonians. It is related to the S-duality of $\cal{N}=4$ super Yang-Mills theory. We propose a one-parameter deformation of the Analytic Langlands Correspondence, and discuss its relations to quantum field theory. The partition functions of the $H_3^+$ WZNW model are interpreted as the wave-functions of a spherical vector in the quantisation of complex Chern-Simons theory. Verlinde line operators generate a representation of two copies of the quantised skein algebra on generalised partition functions. We conjecture that this action generates a basis for the underlying Hilbert space, and explain in which sense the resulting quantum theory represents a deformation of the Analytic Langlands Correspondence.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Published as SciPost Phys. 18, 144 (2025)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-3-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.00494v1, delivered 2025-03-30, doi: 10.21468/SciPost.Report.10932
Strengths
As well as being a pathbreaking paper, this is a very difficult one. A lot of different things go into it. Indeed, I described myself as ``expert,'' but realistically I am only an expert on some aspects of the background and quite a beginner on other aspects. SciPost would have a lot of trouble to find a referee who was really knowledgeable on all of the things that go into this paper.
Weaknesses
The reason that I give the paper ``high'', not ``top'' grades for clarity is that indeed there are places where the explanation of the background could have been more complete.
Report
Requested changes
I don't really want to recommend changes, but for the benefit of the authors, I'd like to mention that the work of Beilinson and Drinfeld to quantize Hitchin's integrable system was undoubtedly stimulated by Hitchin's prior work doing this for SL_2.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by Anonymous (Referee 1) on 2025-2-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.00494v1, delivered 2025-02-10, doi: 10.21468/SciPost.Report.10644
Strengths
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This paper introduces an important and intriguing extension - the quantum deformation - of the analytic Langlands correspondence.
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The physical account of the (quantum) analytic Langlands correspondence is also well-analyzed in 2d/3d/4d perspectives, which would help readers unfamiliar with the subject.
Weaknesses
- While the paper is well-structured, some sections are dedicated to reviewing results from previous works, which extends its length. However, I do not see this as a weakness, as it ensures the paper includes all the necessary details which would eventually help readers.
Report
Requested changes
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It could be helpful to include a figure in Section 4.4.1 or 4.4.2 intuitively showing how the FN coordinates are defined.
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In Section 2.3, the authors discuss the physical account of the Hecke operators. It could be worth mentioning this paper(https://inspirehep.net/literature/2650150) studying the class S theory counterpart.
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In Section 4.6.2, the authors present the twist coordinate $\kappa(\lambda, z)$ for the Heun oper in the limit $z \to 0$ at the order of $\log z$, following [48,49]. It may also be worth mentioning this paper (https://inspirehep.net/literature/1678845) for this small z analysis from the class S theory perspective.
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(6.18) is called the Verlinde line operator insertion in the sense that 1) $H_3 ^+$-WZNW correlation function admits an integral transform presentation (7.9), in which the integrand is given by the Liouville correlation function; 2) The Verline line operators for that Liouville correlation function produces the additional insertion in the integrand of (6.18). Although this is explained well at the beginning of Section 8, it could be helpful to restate this explicitly by adding an equation at the end of Section 8.6.
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Similar to the comment 1, a figure in Section 8.2.2 or 8.2.3 may be helpful.
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As authors summarized in page 17, the analytic Langlands correpsondence involved Hecke operators sharing the eigenspace with Hitchin Hamiltonians. Could there be any comment on the role of Hecke operators after the quantum deformation (or just Hecke modification)?
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In Section 10.1, the authors discuss the higher-rank generalization. It may be worth mentioning this paper (https://inspirehep.net/literature/2760389) studying the higher-rank version of the integral transformation (7.9) in the class S theory perspective.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)