Quantum K-theory of quiver varieties and many-body systems

Article Mathematics

TOROIDAL q-OPERS

Peter Koroteev et al.

Summary: This paper defines and studies the space of q-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. The construction is inspired by the study of enumerative geometry of cyclic quiver varieties, particularly the ADHM moduli spaces. The authors define ((GL) over bar(infinity),q)-opers with regular singularities and then derive the desired Bethe equations for toroidal q-opers by imposing various analytic conditions on singularities.

JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU (2023)

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Article Physics, Mathematical

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

Peter Koroteev et al.

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.

COMMUNICATIONS IN MATHEMATICAL PHYSICS (2021)

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Article Mathematics