ABSTRACT:
In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. A recent generalization of the correspondence, due to Aganagic-Frenkel-Okounkov, establishes an isomorphism between q-deformed versions of conformal blocks on the Riemann surface, for a W-algebra on one side, and a Langlands dual affine Lie algebra on the other side. 
In this talk, we will elucidate the meaning of tame ramification in this correspondence. The crucial new ingredient will be a definition of q-primary vertex operators on the W-algebra side, which we argue to be determined entirely from the representation theory of the dual quantum affine algebra, much like the celebrated relation between W-algebra generators and the q-characters of Frenkel-Reshetikhin. As an application, we propose a construction of fundamental representations of quantum algebras via the supersymmetric Higgs mechanism in gauge theory with 8 supercharges, on an Omega-background.