Communications in Number Theory and Physics
Volume 5 (2011)
Archimedean $L$-factors and topological field theories II
Pages: 101 – 133
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.