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We realize the fundamental representations of quantum algebras via thesupersymmetric Higgs mechanism in gauge theories with 8 supercharges on an$\Omega$-background. We test our proposal for quantum affine algebras, byprobing the Higgs phase of a 5d quiver gauge theory on a circle. We show thatour construction implies the existence of tame ramification in theAganagic-Frenkel-Okounkov formulation of the geometric Langlands program, acorrespondence which identifies $q$-conformal blocks of the quantum affinealgebra with those of a Langlands dual deformed ${\cal W}$-algebra. The newfeature of ramified blocks is their definition in terms of Drinfeld polynomialsfor a set of quantum affine weights. In enumerative geometry, the blocks arevertex functions counting quasimaps to quiver varieties describing modulispaces of vortices. Physically, the vortices admit a description as a 3d ${\calN}=2$ quiver gauge theory on the Higgs branch of the 5d gauge theory, uniquelydetermined from the Drinfeld polynomial data; the blocks are supersymmetricindices for the vortex theory supported on a 3-manifold with distinguished BPSboundary conditions. The top-down explanation of our results is found in the 6d$(2,0)$ little string theory, where tame ramification is provided by certainD-branes. When the string mass is taken to be large, we make contact withvarious physical aspects of the point particle superconformal limit: theGukov-Witten description of ramification via monodromy defects in 4d SuperYang-Mills (and their S-duality), the Nekrasov-Tsymbaliuk solution to theKnizhnik-Zamolodchikov equations, and the classification of massivedeformations of tamely ramified Hitchin systems. In a companion paper, we willshow that our construction implies a solution to the localAlday-Gaiotto-Tachikawa conjecture.

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