A new realization of quantum algebras in gauge theory and Ramification in the Langlands program

Abstract

We realize the fundamental representations of quantum algebras via the supersymmetric Higgs mechanism in gauge theories with 8 supercharges on an $\Omega$-background. We test our proposal for quantum affine algebras, by probing the Higgs phase of a 5d quiver gauge theory on a circle. We show that our construction implies the existence of tame ramification in the Aganagic-Frenkel-Okounkov formulation of the geometric Langlands program, a correspondence which identifies $q$-conformal blocks of the quantum affine algebra with those of a Langlands dual deformed ${\cal W}$-algebra. The new feature of ramified blocks is their definition in terms of Drinfeld polynomials for a set of quantum affine weights. In enumerative geometry, the blocks are vertex functions counting quasimaps to quiver varieties describing moduli spaces of vortices. Physically, the vortices admit a description as a 3d ${\cal N}=2$ quiver gauge theory on the Higgs branch of the 5d gauge theory, uniquely determined from the Drinfeld polynomial data; the blocks are supersymmetric indices for the vortex theory supported on a 3-manifold with distinguished BPS boundary conditions. The top-down explanation of our results is found in the 6d $(2,0)$ little string theory, where tame ramification is provided by certain D-branes. When the string mass is taken to be large, we make contact with various physical aspects of the point particle superconformal limit: the Gukov-Witten description of ramification via monodromy defects in 4d Super Yang-Mills (and their S-duality), the Nekrasov-Tsymbaliuk solution to the Knizhnik-Zamolodchikov equations, and the classification of massive deformations of tamely ramified Hitchin systems. In a companion paper, we will show that our construction implies a solution to the local Alday-Gaiotto-Tachikawa conjecture.