Quantum q-Langlands Correspondence

Abstract

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. The physical origin of the correspondence is the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d string theory becomes the 6d conformal field theory with (2,0) supersymmetry.

116 pages, 4 figures. Minor changes. Version accepted for publication in Trudy Moskovskogo Matematicheskogo Obshchestva. (Transactions of the Moscow Mathematical Society.)

References in corpus (3)

• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• ADE Little String Theory on a Riemann Surface (and Triality)
• Twistorial Topological Strings and a tt* Geometry for N=2 Theories in 4d

Cited by in corpus (9)

• Triality in Little String Theories
• 3d Mirror Symmetry from S-duality
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• Singular Geometry and Higgs Bundles in String Theory
• Bootstrapping the $S^5$ partition function
• Elliptic Weight Functions and Elliptic q-KZ Equation
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Representation theory of W-algebras and Higgs branch conjecture
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians