\(q\)-opers, \(QQ\)-systems, and Bethe Ansatz (Q6192230)

Summary: 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 \(\mathfrak{g}\) 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 \(U_q \hat{\mathfrak{g}}\). However, if \(\mathfrak{g}\) is non-simply-laced, then these equations correspond to a different integrable model, associated to \(U_q {}^L \hat{\mathfrak{g}}\) where \({}^L \hat{\mathfrak{g}}\) 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 \(\mathcal{O}\) of the relevant quantum affine algebra.