The Gaudin model in the Deligne category Rep $GL_t$

Abstract

Deligne's category $D_t$ is a formal way to define the category of finite-dimensional representations of the group $GL_n$ with $n=t$ being a formal parameter (which can be specialized to any complex number). I will show how to interpolate the construction of the higher Hamiltonians of the Gaudin quantum spin chain associated with the Lie algebra $\mathfrak{gl}_n$ to any complex $n$, using $D_t$. Next, according to Feigin and Frenkel, Bethe ansatz equations in the Gaudin model are equivalent to no-monodromy conditions on a certain space of differential operators of order $n$ on the projective line. We also obtain interpolations of these no-monodromy conditions to any complex $n$ and prove that they generate the relations in the algebra of higher Gaudin Hamiltonians for generic complex $n$. I will also explain how it is related to the Bethe ansatz for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{m|n}$.   This is joint work with Boris Feigin and Filipp Uvarov, https://arxiv.org/abs/2304.04501.