Regularization and Renormalization of Chern-Simons Theory

TL;DR

This paper investigates the perturbative quantization of Chern-Simons theory, demonstrating its finiteness in Landau gauge, analyzing renormalization schemes, and showing that certain regularizations preserve observable invariance.

Contribution

It introduces a natural renormalization scheme where the classical parameter remains unchanged and shows the invariance of Wilson loop observables across different regularizations.

Findings

Chern-Simons theory is perturbatively finite in Landau gauge.

Wilson loop observables depend only on shifted monodromy parameter across regulators.

Two-loop corrections can be absorbed by field rescaling, suggesting higher-order invariance.

Abstract

We analyze some features of the perturbative quantization of Chern-Simons theory (CST) in the Landau gauge. In this gauge the theory is known to be perturbatively finite. We consider the renormalization scheme in which the renormalized parameter $k$ equals the bare or classical one and show that it constitutes a natural parametrization for the quantum theory. The reason is that, although in this renormalization scheme the value of the Green functions depends on the regularization used, comparison among different regularization methods shows that the observables (Wilson loops) are the same function of the shifted monodromy parameter $k+c_v$ for all BRS invariant regulators used so far for CST. We also discuss a particular BRS invariant regularization prescription in which CST is perturbatively defined as the large mass limit of dimensionally regularized topologically massive Yang-Mills theory. With this regularization prescription the radiative corrections induced by two-loop contributions do not entail observable consequences since they can be reabsorbed by a finite rescaling of the fields only. This very mechanism is conjectured to take place at higher perturbative orders. Talk presented by G.G. at the NATO AWR on ``Low dimensional Topology and Quantum Field Theory'', 6-13 September 1992, Cambridge (UK).