Archimedean $L$-factors and topological field theories II

Abstract

In the first part of this series of papers we propose a functional integral representation for local Archimedean L-factors given by products of the Gamma-functions. In particular we derive a representation of the Gamma-function as a properly regularized equivariant symplectic volume of an infinite-dimensional space. The corresponding functional integral arises in the description of a type A equivariant topological linear sigma model on a disk. In this paper we provide a functional integral representation of the Archimedean L-factors in terms of a type B topological sigma model on a disk. This representation leads naturally to the classical Euler integral representation of the Gamma-functions. These two integral representations of L-factors in terms of A and B topological sigma models are related by a mirror map. The mirror symmetry in our setting should be considered as a local Archimedean Langlands correspondence between two constructions of local Archimedean L-factors.