Renormalization and finiteness of topological BF theories

TL;DR

This paper demonstrates that topological BF theories in any dimension are ultraviolet finite due to an underlying superalgebra structure, with no anomalies or counterterms arising in certain gauges.

Contribution

It provides a general algebraic proof of the finiteness and anomaly absence of topological BF theories across dimensions, highlighting their superalgebra symmetry.

Findings

BF theories possess vector supersymmetry in certain gauges

Ultraviolet finiteness is proven algebraically in lower dimensions

No anomalies or counterterms are found in the analyzed cases

Abstract

We show that the BF theory in any space-time dimension, when quantized in a certain linear covariant gauge, possesses a vector supersymmetry. The generator of the latter together with those of the BRS transformations and of the translations form the basis of a superalgebra of the Wess-Zumino type. We give a general classification of all possible anomalies and invariant counterterms. Their absence, which amounts to ultraviolet finiteness, follows from purely algebraic arguments in the lower-dimensional cases.