Quantum~$q$-Langlands correspondence

M. Aganagic ; E. Frenkel ; A. Okounkov

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 1, pp. 1-95

Résumé

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 algebra 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 $\mathrm{K}$-theory of the Nakajima quiver varieties. The physical origin of the isomorphism between them lies in the $\mathrm{6d}$ little string theory. The quantum Langlands correspondence emerges in the limit in which the $\mathrm{6d}$ little string theory becomes the $\mathrm{6d}$ conformal field theory with $(2,0)$ supersymmetry. References: 130 entries.