Abstract: Integrable models are known to keep reemerging over time in various mathematical incarnations. Recently, such models based on quantum groups naturally appeared in the framework of enumerative geometry. In this context the so-called Bethe ansatz equations, instrumental for finding the spectrum of the XXZ model Hamiltonian, naturally show up as constraints for the quantum K-theory ring of quiver varieties.

In this talk I will describe another geometric interpretation of Bethe ansatz equations, which is indirectly related to the above. I will introduce the notion of (G,q)-opers, the difference analogue of oper connections for simply connected group G. I will explain the one-to-one correspondence between (G,q)-opers of specific kind and Bethe equations for XXZ models. The key element in this identification is the so-called QQ-system, which has previously appeared in the study of ODE/IM correspondence and the Grothendieck ring of the category O of the relevant quantum algebras. I will speculate on how that fits into recently proposed quantum q-Langlands correspondence by M. Aganаgic, E. Frenkel and A. Okounkov.

The talk is based on joint work with E. Frenkel, P. Koroteev and D. Sage ( arXiv:1811.09937, arXiv:2002.07344)