q-opers, QQ-systems, and Bethe Ansatz

Abstract

We introduce the notions of .G; q/-opers and Miura .G; q/-opers, where G is a simply connected simple complex Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of .G; q/-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a qDE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects (q-differential equations). If g is simply laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra Uqyg. However, if g is non-simply-laced, then these equations correspond to a different integrable model, associated to Uq Lyg where Lyg is the Langlands dual (twisted) affine algebra. A key element in this qDE/IM correspondence is the QQ-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category O of the relevant quantum affine algebra.