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(glm, gln)-dualities in gaudin models with irregular singularities

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Original languageEnglish
Article number040
Number of pages28
JournalSIGMA
Journal publication date3 May 2018
Volume14
DOIs
Publication statusPublished - 3 May 2018

Abstract

We establish (gl M, gl N)-dualities between quantum Gaudin models with irregular singularities. Specifically, for any M,N ∈ ℤ ≥1 we consider two Gaudin models: the one associated with the Lie algebra gl M which has a double pole at infinity and N poles, counting multiplicities, in the complex plane, and the same model but with the roles of M and N interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization we consider the classical cyclotomic Gaudin model associated with the Lie algebra gl M and its diagram automorphism, with a double pole at infinity and 2N poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra sp 2N, with a double pole at infinity and M simple poles in the complex plane. In the special case N=1 we recover the well-known self-duality in the Neumann model.

Notes

This document is the Accepted Manuscript version of the following article: Benoit Vicedo, and Charles Young, ‘Dualities in Gaudin Models with Irregular Singularities’, SIGMA, Vol. 14 (040), May 2018. The final, definitive version is available online via: https://doi.org/10.3842/SIGMA.2018.040 Content in the UH Research Archive is made available for personal research, educational, and non-commercial purposes only. Unless otherwise stated, all content is protected by copyright, and in the absence of an open license, permissions for further re-use should be sought from the publisher, the author, or other copyright holder.

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