The consistency of topological expansions in field theory : “BRST anomalies” in strings and Yang-Mills.

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 cutoff 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 counterterm in the former case, and by compactification on S4 in the latter