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\(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality

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Abstract

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for \({{\,\mathrm{\mathrm {SL}}\,}}(N)\). We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars–Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties.

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Notes

  1. We remark that the Gaudin algebra is a subalgebra of the center of the universal enveloping algebra at the critical level, and this center was characterized in [FFR1]. Ding and Etingof have found generators of the center of the quantum affine algebra at the critical level, thereby proving the q-analogue of this statement [DE].

  2. The authors and E. Frenkel have subsequently solved this problem in the general case using different techniques [FKSZ].

  3. We refer the reader to Sect. 4 of [GK] for more information on quantum/classical duality and additional references.

  4. The terminology comes from the analogy between these polynomials and the polynomials determining the eigenvalues of the Baxter operators arising from transfer matrices [R2].

  5. The quantum K-theory discussed here is the so-called PSZ quantum K-theory [PSZ], involving quasimaps to Nakajima varieties, as opposed to the Givental–Lee approach which relies on stable maps. We refer to [SZ] for some recent progress in comparing the two approaches.

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Acknowledgements

P.K. and A.M.Z. are grateful to the 2018 Simons Summer Workshop for providing a wonderful working atmosphere in the early stages of this Project. P.K. thanks the organizers of the program “Exactly Solvable Models of Quantum Field Theory and Statistical Mechanics” at the Simons Center for Geometry and Physics, where part of this work was done. We are also indebted to N. Nekrasov, A. Schwarz, and Y. Soibelman for stimulating discussions and suggestions. D.S.S. was partially supported by NSF Grant DMS-1503555 and Simons Collaboration Grant No. 637367, and A.M.Z. was partially supported by Simons Collaboration Grant No. 578501. P.K. is partly supported by an AMS-Simons Grant.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Peter Koroteev

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Daniel S. Sage & Anton M. Zeitlin

  3. IPME RAS, St. Petersburg, Russia

    Anton M. Zeitlin

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  1. Peter Koroteev
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Correspondence to Anton M. Zeitlin.

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Communicated by Y. Kawahigashi

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Koroteev, P., Sage, D.S. & Zeitlin, A.M. \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality. Commun. Math. Phys. 381, 641–672 (2021). https://doi.org/10.1007/s00220-020-03891-1

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