Folded quantum integrable models and deformed W-algebras

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.Comment: 70 pages; v2: added a reference and Remarks 1.1 and 9.2; v3: minor changes, accepted for publication in Letters in Mathematical Physic