Quantum $q$-Langlands Correspondence

Abstract:

We conjecture, and prove for all simply-laced Lie algebras, an identification between the spaces of $q$-deformed conformal blocks for the deformed ${\mathcal W}$-algebra ${\mathcal W}_{q,t}(\mathfrak {g})$ and quantum affine algebras of $\widehat {^L\mathfrak {g}}$, where $^L\mathfrak {g}$ is the Langlands dual Lie algebra to $\mathfrak {g}$. We argue that this identification may be viewed as a manifestation of a $q$-deformation of the quantum Langlands correspondence. Our proof relies on expressing the $q$-deformed conformal blocks for both algebras in terms of the quantum K-theory of the Nakajima quiver varieties. The physical origin of the isomorphism between them lies in the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d little string theory becomes the 6d conformal field theory with $(2,0)$ supersymmetry.