The Consistency of Topological Expansions in Field Theory: `BRST Anomalies' in Strings and Yang-Mills

TL;DR

This paper investigates the topological expansion in field theories like strings and Yang-Mills, focusing on BRST anomalies and methods to remove parametrisation dependence through counter-terms and compactification.

Contribution

It analyzes the consistency of topological expansions in field theories and proposes specific techniques to address BRST anomalies and divergences.

Findings

Counter-term removal in Bosonic Strings

Compactification on S^4 in Yang-Mills

Resolution of parametrisation dependence

Abstract

Many field theories of physical interest have configuration spaces consisting of disconnected components. Quantum mechanical amplitudes are then expressed as sums over these components. We use the Faddeev-Popov approach to write the terms in this topological expansion as moduli space integrals. A cut-off is needed when these integrals diverge. This introduces a dependence on the choice of parametrisation of configuration space which must be removed if the theory is to make physical sense. For theories that have a local symmetry this also leads to a breakdown in BRST invariance. We discuss in detail the cases of Bosonic Strings and Yang-Mills theory, showing how this arbitrariness may be removed by the use of a counter-term in the former case, and by compactification on $S\sp 4$ in the latter.