Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
Wan Keng Cheong, Ngau Lam
TL;DR
The paper proves a duality between Gaudin algebras with irregular singularities for ${\mathfrak{gl}}_d$ and ${\mathfrak{gl}}_{p+m|q+n}$ acting on a shared Fock space, realized through explicit neuro-algebra homomorphisms ${\phi}$ and ${\varphi}$ into the Weyl superalgebra. The main result equates the actions ${\mathcal A}_d^{\mathbf{w}, \boldsymbol{\xi}}(\mathbf z, \boldsymbol{\gamma})$ and ${\mathcal A}_{p+m|q+n}^{\mathbf z, \boldsymbol{\gamma}}(\mathbf w, \boldsymbol{\xi})$, yielding a duality for Gaudin models that extends prior bosonic results to the super setting. The authors provide an application showing cyclicity and simple spectrum on weight spaces of classes of infinite-dimensional Takiff modules, and they establish a classical analogue, recovering Vicedo–Young dualities as special cases. Overall, the work unifies dualities across bosonic and fermionic realizations with irregular singularities and opens pathways for representation-theoretic and integrable-systems applications.
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.