Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing

TL;DR

This paper investigates the perturbative quantization of 2D massive scalar field theory with polynomial potentials, demonstrating its compatibility with Atiyah-Segal functorial quantum field theory and establishing a gluing formula involving tadpoles.

Contribution

It establishes that perturbative 2D scalar QFT with polynomial potentials fits into the Atiyah-Segal framework, including a novel gluing formula accounting for tadpole contributions.

Findings

Partition function satisfies Atiyah-Segal gluing formula.

Tadpoles (short loops) are crucial in the gluing process.

The framework applies to manifolds with boundary.

Abstract

We study the perturbative quantization of 2-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorial quantum field theory framework of Atiyah-Segal. In particular, we prove that the perturbative partition function defined in terms of integrals over configuration spaces of points on the surface satisfies an Atiyah-Segal type gluing formula. Tadpoles (short loops) behave nontrivially under gluing and play a crucial role in the result.