$$({{\,\mathrm{\mathrm {SL}}\,}}(N),q)$$ ( SL
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Communications in
Mathematical Physics
(SL(N), q)-Opers, the q -Langlands Correspondence, and Quantum/Classical Duality Peter Koroteev1 , Daniel S. Sage2 , Anton M. Zeitlin2,3 1 Department of Mathematics, University of California, Berkeley, CA 94720, USA.
E-mail: [email protected]
2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA.
E-mail: [email protected]
3 IPME RAS, St. Petersburg, Russia. E-mail: [email protected]
Received: 25 July 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 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. Contents 1.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Opers and the Gaudin model . . . . . . . . . . . . . . . . . . . . . 1.2 q-opers and the q-Langlands correspondence . . . . . . . . . . . . 1.3 QQ-systems, XXZ models, and Baxter operators . . . . . . . . . . . 1.4 Quantum/classical duality and applications to enumerative geometry 1.5 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . SL(N )-Opers with Trivial Monodromy and Regular Singularities . . . . 2.1 SL(2)-opers and Bethe equations . . . . . . . . . . . . . . . . . . . 2.2 Miura opers and the Miura transformation . . . . . . . . . . . . . . 2.3 Generalization to SL(N ): a brief summary . . . . . . . . . . . . . . 2.4 Irregular singularities . . . . . . . . . . . . . . . . . . . . . . . . . (SL(2), q)-Opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The quantum Wronskian and the Bethe ansatz . . . . . . . . . . . .
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P. Koroteev, D. S. Sage, A. M. Zeitlin
3.3 The q-Miura transformation and the transfer matrix . . . . . . . 3.4 Embedding of the tRS model into q-opers . . . . . . . . . . . . 4. (SL(N ), q)-Opers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Miura q-opers and quantum Wronskians . . . . . . . . . . . . . 4.3 (SL(N ), q)-Opers and the XXZ Bethe ansatz . . . . . . . . . . 5. Explicit Equations for (SL(3), q)-Opers
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