Folded quantum integrable models and deformed W-algebras

TL;DR

This paper introduces a new class of quantum integrable models for non-simply laced Lie algebras, connecting their spectra to folded Bethe Ansatz equations and deformed W-algebras, with implications for representation theory and Langlands duality.

Contribution

It constructs folded quantum integrable models for all non-simply laced Lie algebras, linking their spectra to folded Bethe Ansatz equations and deformed W-algebras, and explores their representation-theoretic implications.

Findings

Spectra correspond to solutions of folded Bethe Ansatz equations.

Spaces of states are identified with representations of the Langlands dual quantum affine algebra.

Conjectural construction of g-crystals via folded q-characters.

Abstract

We propose a novel quantum integrable model for every non-simply 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 $U_q(\hat{\mathfrak g'})$ of the simply-laced 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 finite-dimensional representations of $U_q(\hat{\mathfrak g'})$. We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of $U_q({}^L\hat{\mathfrak g})$, where $^L\hat{\mathfrak g}$ is the (twisted) affine Kac-Moody algebra Langlands dual to $\hat{\mathfrak g}$. 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.