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.

    Source link


    Leave A Reply