Gaudin model and Deligne’s category
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Author(s)
Feigin, B.
Rybnikov, L.
Uvarov, F.
Date Issued
December 19, 2023
Publisher
Springer Netherlands
Citation
Letters in Mathematical Physics. 2023 Dec 19;114(1):3
Version
Author's final manuscript
Abstract
We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra
$$\mathfrak {gl}_{n}$$
gl
n
admits an interpolation to any complex number n. We do this using the Deligne’s category
$$\mathcal {D}_{t}$$
D
t
, which is a formal way to define the category of finite-dimensional representations of the group
$$GL_{n}$$
G
L
n
, when n is not necessarily a natural number. We also obtain interpolations to any complex number n of the no-monodromy conditions on a space of differential operators of order n, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex n are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-differential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra
$$\mathfrak {gl}_{n\vert n'}$$
gl
n
|
n
′
, we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11005-023-01747-y
