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.

💡 Research Summary

The paper establishes a new duality between Gaudin algebras with irregular singularities associated to the general linear Lie algebra gl₍d₎ and the general linear Lie superalgebra gl₍p+m|q+n₎. The authors consider sequences of complex points w = (w₁,…,w_{d′}) and z = (z₁,…,z_ℓ) together with integer multiplicities ξ = (ξ₁,…,ξ_{d′}) and γ = (γ₁,…,γ_ℓ). For each point they form Takiff algebras gl₍d₎(z,γ) = ⊕_{i=1}^ℓ gl₍d₎