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

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.

Automated summary

This paper discusses a deformation of the geometric Langlands correspondence for SL(N) and introduces the concept of q-opers. It proves 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. It also suggests that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model can be seen as a special case of the q-Langlands correspondence, and explores the application of q-opers to equivariant quantum K-theory of cotangent bundles to partial flag varieties.