Archimedean L-factors and Topological Field Theories II

TL;DR

This paper develops a novel functional integral representation of Archimedean L-factors using topological sigma models of types A and B, revealing a mirror symmetry that acts as a local Archimedean Langlands correspondence.

Contribution

It introduces new functional integral representations of Archimedean L-factors via topological sigma models and establishes a mirror symmetry linking these models as a form of local Langlands correspondence.

Findings

Functional integral representation of Gamma-functions as equivariant symplectic volumes.

Representation of L-factors using type B topological sigma models.

Identification of mirror symmetry as a local Archimedean Langlands correspondence.

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.