$({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities

We establish $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-dualities between quantum Gaudin models with irregular singularities. Specifically, for any $M, N \in {\mathbb Z}_{\geq 1}$ we consider two Gaudin models: the one associated with the Lie algebra ${\mathfrak{gl}}_M$ which has a double pole at infinity and $N$ poles, counting multiplicities, in the complex plane, and the same model but with the roles of $M$ and $N$ interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization we consider the classical cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{gl}}_M$ and its diagram automorphism, with a double pole at infinity and $2N$ poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{sp}}_{2N}$, with a double pole at infinity and $M$ simple poles in the complex plane. In the special case $N=1$ we recover the well-known self-duality in the Neumann model.