Deligne's category, monodromy-free pseudo-differential operators and Gaudin model for the Lie superalgebra $gl(m|n)$.

Abstract: The Deligne's category is a formal way to define an interpolation of the category of finite-dimensional representations of the Lie group $GL(n)$ to any complex number $n$. It is used in various constructions, which all together can be named as representation theory in complex rank. In the talk, I will present one of such constructions, namely, the interpolation of the algebra of higher Gaudin Hamiltonians (the Bethe algebra) associated with the Lie algebra $gl(n)$. One can also interpolate monodromy-free differential operators of order $n$ desribing eigenvectors of Gaudin Hamiltonians, obtaining "monodromy-free" pseudo-differential operators. In joint work with L. Rybnikov and B. Feigin arXiv:2304.04501, we prove that the Bethe algebra in Deligne's category is isomorphic to the algebra of functions on certain pseudo-differential operators. Our work is motivated by the Bethe ansatz conjecture for the case of Lie superalgebra $gl(m|n)$. The conjecture says that eigenvectors in this case are described by ratios of differential operators of orders $m$ and $n$. We prove that such ratios are "monodromy-free" pseudo-differential operators.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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