From Archimedean L-Factors to Topological Field Theories

Abstract

We give a short account of our recent results on establishing a relation between topological field theories and Archimedean (∞-adic) geometry. First, we introduce a universal integral Baxter operator as a certain element of the spherical Hecke algebra \({\fancyscript{H}(G,K)}\). Eigenvalues of the Baxter operator acting on spherical vectors in irreducible representations G are given by corresponding local Archimedean L-factors. We provide a quantum field theory interpretation of the local Archimedean L-factors as certain correlation functions in a two-dimensional type A equivariant topological linear sigma model on a disk D. Next, we introduce a q-version of the Archimedean L-factors and construct a particular representation of q-deformed Whittaker functions generalizing the Casselman–Shalika formula for p-adic class one Whittaker functions. This representation allows to formulate a q-version of Archimedean Langlands correspondence. Then, we construct a representation of q-version of the Archimedean L-factors as a certain correlation functions in three-dimensional equivariant linear sigma model on D × S1. Finally, we propose an interpretation of the Archimedean Langlands correspondence as a mirror symmetry in the underlying topological quantum field theory.