TY - JOUR

T1 - (glm, gln)-dualities in gaudin models with irregular singularities

AU - Vicedo, Benoit

AU - Young, Charles

N1 - © 2018 The Author(s). This article is made available under the terms of the Creative Commons Attribution-ShareAlike License (https://creativecommons.org/licenses/by-sa/4.0/)

PY - 2018/5/3

Y1 - 2018/5/3

N2 - 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.

AB - 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.

KW - Dualities

KW - Gaudin models

KW - Irregular singularities

UR - http://www.scopus.com/inward/record.url?scp=85046539504&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2018.040

DO - 10.3842/SIGMA.2018.040

M3 - Article

SN - 1815-0659

VL - 14

JO - SIGMA

JF - SIGMA

M1 - 040

ER -