Communications in Number Theory and Physics

Volume 5 (2011)

Number 1

Archimedean $L$-factors and topological field theories II

Pages: 101 – 133



Anton Gerasimov (Institute for Theoretical and Experimental Physics, Moscow, Russia)

Dimitri Lebedev (School of Mathematics, Trinity College, Dublin, Ireland)

Sergey Oblezin (Hamilton Mathematics Institute, Trinity College, Dublin, Ireland)


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.

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