Opers on the projective line, Wronskian relations, and the Bethe Ansatz

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate $G$-opers: $G$-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of $G$. They exhibited the Bethe Ansatz equations for this model and gave a geometric interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Pl\"ucker opers and a certain system of nonlinear differential equations called the $qq$-system. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the $qq$-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Pl\"ucker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Pl\"ucker opers are in fact Miura opers.