Folded quantum integrable models and deformed Walgebras
Abstract
We propose a novel quantum integrable model for every nonsimply laced simple Lie algebra ${\mathfrak g}$, which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated to the quantum affine algebra $U_q(\widehat{\mathfrak g'})$ of the simplylaced Lie algebra ${\mathfrak g}'$ corresponding to ${\mathfrak g}$. Our construction is motivated by the analysis of the second classical limit of the deformed ${\mathcal W}$algebra of ${\mathfrak g}$, which we interpret as a "folding" of the Grothendieck ring of finitedimensional representations of $U_q(\widehat{\mathfrak g'})$. We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finitedimensional representations of $U_q({}^L\widehat{\mathfrak g})$, where $^L\widehat{\mathfrak g}$ is the (twisted) affine KacMoody algebra Langlands dual to $\widehat{\mathfrak g}$. We discuss the analogous structures in the Gaudin model which appears in the limit $q \to 1$. Finally, we describe a conjectural construction of the simple ${\mathfrak g}$crystals in terms of the folded $q$characters.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.14600
 Bibcode:
 2021arXiv211014600F
 Keywords:

 Mathematics  Quantum Algebra;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 69 pages, v2: added a reference and Remarks 1.1 and 9.2