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# Communications in Number Theory and Physics

## Volume 5 (2011)

### Number 1

### Archimedean $L$-factors and topological field theories II

Pages: 101 – 133

DOI: https://dx.doi.org/10.4310/CNTP.2011.v5.n1.a3

#### Authors

#### Abstract

In the first part of this series ofpapers, we propose a functional integral representation for localArchimedean $L$-factors given by products of the$\Gamma$-functions. In particular, we derive a representation of the$\Gamma$-function as a properly regularized equivariant symplecticvolume of an infinite-dimensional space. The correspondingfunctional 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\break$L$-factors in terms of a type $B$ topological sigma model on adisk. This representation leads naturally to the classicalEuler integral representation of the $\Gamma$-functions. These twointegral representations of $L$-factors in terms of $A$ and $B$topological sigma models are related by a mirror map. The mirrorsymmetry in our setting should be considered as a local ArchimedeanLanglands correspondence between two constructions of localArchimedean $L$-factors.

Published 2 September 2011