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# Communications in Number Theory and Physics

## Volume 5 (2011)

### Number 1

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

Pages: 57 – 100

DOI: https://dx.doi.org/10.4310/CNTP.2011.v5.n1.a2

#### Authors

#### Abstract

We propose a functional integral representation for Archimedean $L$-factors given by products of $\Gamma$-functions.The corresponding\break functional integral arises in the description of type-A equivariant topological linear sigma modelon a disk. The functional integral representation provides in particular an interpretation of\break the $\Gamma$-function asan equivariant symplectic volume of an infinite-\break dimensional space of holomorphic maps of the disk to $\IC$. Thisshould be considered as a mirror dual to the classical Euler integral representation of the $\Gamma$-function. We givean analogous functional integral representation of $q$-deformed $\Gamma$-functions using a three-dimensionalequivariant topological linear sigma model on a handlebody. In general, upon proper ultra violent completion, thetopological sigma model considered on a particular class of three-dimensional spaces with a compact Kähler targetspace provides a quantum field theory description of a $K$-theory version of Gromov–Witten invariants.

Published 2 September 2011