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BIMSA Integrable Systems Seminar
Deligne's category, monodromy-free pseudo-differential operators and Gaudin model for the Lie superalgebra $gl(m|n)$
Deligne's category, monodromy-free pseudo-differential operators and Gaudin model for the Lie superalgebra $gl(m|n)$
演讲者
Filipp Uvarov
时间
2024年10月29日 16:00 至 17:00
地点
A6-101
线上
Zoom 873 9209 0711
(BIMSA)
摘要
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.
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.