Archimedean L-factors and Topological Field Theories I

TL;DR

This paper introduces a novel functional integral representation for Archimedean L-factors, linking Gamma-functions to topological field theories and symplectic geometry, and extends it to q-deformed cases within a quantum field theory framework.

Contribution

It provides a new functional integral approach to Archimedean L-factors and their q-deformations using topological sigma models, offering a geometric and physical interpretation.

Findings

Gamma-functions as equivariant symplectic volumes

Functional integral representation of q-deformed Gamma-functions

Topological sigma models describe K-theoretic Gromov-Witten invariants

Abstract

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