Chern-Simons Perturbation Theory

TL;DR

This paper develops a perturbation theory for 3D Chern-Simons quantum field theory, demonstrating finiteness and addressing metric dependence issues through a local counterterm aligned with the theory's framing dependence.

Contribution

It introduces a superspace formulation of the gauge-fixed action, analyzes propagator singularities, and proves the finiteness of the theory along with a method to cancel metric dependence at two loops.

Findings

The theory is finite after gauge fixing.

A local counterterm cancels metric dependence of the 2-loop partition function.

The counterterm matches the Chern-Simons action of the metric connection.

Abstract

We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the $2$-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the $1$-loop case. In fact, the counterterm is equal to the Chern--Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten's exact solution.