Abstract
We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra \(\mathfrak {gl}_{n}\) admits an interpolation to any complex number n. We do this using the Deligne’s category \(\mathcal {D}_{t}\), which is a formal way to define the category of finite-dimensional representations of the group \(GL_{n}\), 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 \(\mathfrak {gl}_{n\vert n'}\), we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.
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Acknowledgements
We are grateful to the referees for the careful reading of the text and for many valuable remarks. The work was accomplished during L.R.’s stay at the Institut des Hautes Études Scientifiques (IHÉS) and at Harvard University. L.R. would like to thank IHÉS, especially Maxim Kontsevich, and Harvard University, especially Dennis Gaitsgory, for their hospitality.
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The work of F. U. is an output of a research project implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE University).
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Feigin, B., Rybnikov, L. & Uvarov, F. Gaudin model and Deligne’s category. Lett Math Phys 114, 3 (2024). https://doi.org/10.1007/s11005-023-01747-y
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DOI: https://doi.org/10.1007/s11005-023-01747-y