String-Math Seminar: q-Opers, q-Langalnds and Classical/Quantum duality

Seminar | January 28 | 2-3 p.m. | 402 LeConte Hall

 Peter Koroteev, UC Berkeley

 Department of Mathematics

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. 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.