  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.
