Folded quantum integrable models and deformed W-algebras

Abstract

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra 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 Uq( g ′ ) of the simply-laced Lie algebra g ′ corresponding to g. Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of g, which we interpret as a "folding" of the Grothendieck ring of finite-dimensional representations of Uq( g ′ ). We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of Uq( L g), where L g is the (twisted) affine Kac-Moody algebra Langlands dual to g. We discuss the analogous structures in the Gaudin model which appears in the limit q → 1. Finally, we describe a conjectural construction of the simple g-crystals in terms of the folded q-characters. Contents 1. Introduction 1 2. Notation and setup 8 3. Deformed W-algebras and screening operators 10 4. Classical limits of the deformed W-algebra and q-characters 15 5. Folded integrable models 20 6. Interpolating (q, t)-characters 33 7. Examples 43 8. Connection to monomial crystals 49 9. The Gaudin model 52 10. Appendix: Toward constructing a folded version of the qKZ equations for non-simply laced Lie algebras 64 References 66

The $(q,t)$-Cartan matrix specialized at $q=1$

“…Remark. When the authors almost finish the first draft of this paper, Frenkel-Hernandez-Reshetikhin uploaded the paper [7] at Arxiv, which looks deeply related to this present paper. They also consider the virtual quantum Grothendieck ring K q (they use the notation K − t (g)) and observe the "folded" phenomenon by using (q, t)-characters and the their specialization at q = 1.…”

“…Remark 5.9. Frenkel-Hernandez-Reshetikhin also introduced K q in [7] (they use the notation K − t (g)) very recently in the context of "folded" phenomenon via (q, t)-characters and their specializations at q = 1. It can be also understood as an intersection of quantized screening operators S i,q (i ∈ △ 0 ) on X q .…”

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

“…Recall that an irreducible V ∈ U q Lg -mod fd is real if V ⊗ V is irreducible. For example, fundamental modules and Kirillov-Reshetikhin modules are real [Kas02,CH10,FHR21]. From the unitarity and non-vanishing conditions, one obtains the following easy corollary (e.g., [FHR21,Lem.…”

$Q\widetilde{Q}$-systems for twisted quantum affine algebras

“…It is worth to remark that there are other types of relations discovered in the Grothendieck ring. For example, the QQ * -systems in [15, Section 3.4] and the folded QQ-systems in [16,Section 5.7]. They are different from the Q Q-systems of twisted quantum affine algebras considered in this paper.…”

“…We give a detailed proof of a refined version of formulas of coproducts on Drinfeld generators, which are needed in the construction of asymptotic representations, and also need as we noted in Remark 2. 16. We construct asymptotic representations of Borel subalgebras following the method in [23].…”