The Gaudin model in the Deligne category Rep GLtConfirmed

by Leonid Rybnikov (Université de Montreal)

Thursday Nov 28, 2024, 11:00 a.m. → 12:30 p.m. America/Toronto

PI/4-400 - Space Room (Perimeter Institute for Theoretical Physics)

Deligne's category $D_t$ is a formal way to define the category of finite-dimensional representations of the group $G{L}_n$ 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.