Journal Article

q$-opers, $QQ$-systems, and Bethe Ansatz

Zeitlin, Anton M., Sage, Daniel, Frenkel, Edward, Koroteev, Peter

2024

Overview

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 q DE/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 U_q g . However, if g is non-simply-laced, then these equations correspond to a different integrable model, associated to U_q ^Lg where ^L g is the Langlands dual (twisted) affine algebra. A key element in this q DE/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.

• Publisher: European Mathematical Society (EMS)
• Subject: Algebra / Lie groups