We present a gauge-theoretic interpretation of the “analytic” version geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete acting on Hilbert space quantum states. The gauge theory ingredients required to understand this construction—such electric–magnetic duality between Wilson ’t Hooft line four-dimensional theory—are same ones that enter unders...

We introduce a $C^*$-algebra $\mathscr{A}_V$ of variety $V$ over the number field $K$ and $\mathscr{A}_G$ reductive group $G$ ring adeles $K$. Using Pimsner's Theorem we construct an embedding $\mathscr{A}_V\hookrightarrow \mathscr{A}_G$, where is $G$-coherent variety, e.g. Shimura $G$. The analog Langlands reciprocity for $C^*$-algebras. It follows from $K$-theory inclusion $\mathscr{A}_V\subs...

Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also ...

This is an attempt to formulate a geometric Langlands conjecture for G = GSpin2n+1 and GSp2n, n ≥ 1. Namely, let Ǧ be the Langlands dual group (over Q̄l), let X be a smooth projective connected curve. Given a Ǧ-local system E on X , assuming some irreducibility type conditions on VE for some representations V of Ǧ, we propose a conjectural construction of a distinguished E-Hecke automorphic shea...