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

TL;DR

This paper establishes a duality between Gaudin models with irregular singularities for general linear Lie (super)algebras, revealing new symmetries and spectral properties in their representations and classical counterparts.

Contribution

It proves a duality between actions of Gaudin algebras with irregular singularities for $rak{gl}_d$ and $rak{gl}_{p+m|q+n}$ on Fock spaces, extending understanding of their symmetries.

Findings

Gaudin algebra acts cyclically on certain modules

Action is diagonalizable with simple spectrum under generic conditions

Duality holds for classical Gaudin models

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.