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

Abstract

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.

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• Gaudin models with irregular singularities
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