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.
Publication Source (Journal or Book title)
Communications in Mathematical Physics
Koroteev, P., Sage, D., & Zeitlin, A. (2021). (SL(N),q) -Opers, the q-Langlands Correspondence, and Quantum/Classical Duality. Communications in Mathematical Physics, 381 (2), 641-672. https://doi.org/10.1007/s00220-020-03891-1