The Gaudin model in the Deligne category Rep $GL_t$

          @misc{ scivideos_PIRSA:24110087,
            doi = {10.48660/24110087},
            url = {https://pirsa.org/24110087},
            author = {},
            keywords = {Mathematical physics},
            language = {en},
            title = {The Gaudin model in the Deligne category Rep $GL_t$},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2024},
            month = {nov},
            note = {PIRSA:24110087 see, \url{https://scivideos.org/index.php/pirsa/24110087}}
          }
          

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.