Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for $\mathfrak{gl}_{p+m|q+n}$ acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over $\mathfrak{gl}_{p+m|q+n}$ and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for classical Gaudin models.