PIRSA:24110087 Scientific Series Mathematical physics

DOI10.48660/24110087

Series: Mathematical Physics

(2024). The Gaudin model in the Deligne category Rep $GL_t$. Perimeter Institute. https://pirsa.org/24110087

The Gaudin model in the Deligne category Rep $GL_t$. Perimeter Institute, Nov. 28, 2024, https://pirsa.org/24110087 

@misc{ pirsa_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},
  year = {2024},
  month = {nov},
  note = {PIRSA:24110087 see, \url{https://pirsa.org}}
}

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.

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