q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors

Abstract: In this paper, we describe a certain kind of $q$-connections on a projective line, namely $Z$-twisted $(G,q)$-opers with regular singularities using the language of generalized minors. In part one arXiv:2002.07344 we explored the correspondence between these $q$-connections and $QQ$-systems/Bethe Ansatz equations. Here we associate to a $Z$-twisted $(G,q)$-oper a class of meromorphic sections of a $G$-bundle, satisfying certain difference equations, which we refer to as generalized $q$-Wronskians. Among other things, we show that the $QQ$-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.