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arXiv · 1811.09937

$(SL(N),q)$-opers, the $q$-Langlands correspondence, and quantum/classical duality

Peter Koroteev·Daniel S. Sage·Anton M. Zeitlin
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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 $SL(N)$. 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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BibTeXRIS

Peter Koroteev, Daniel S. Sage, Anton M. Zeitlin. 2018-11-25. $(SL(N),q)$-opers, the $q$-Langlands correspondence, and quantum/classical duality. https://doi.org/10.1007/s00220-020-03891-1

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