Local Anomalies, Local Equivariant Cohomology and the Variational Bicomplex

TL;DR

This paper explores the geometric and cohomological structures underlying local anomalies in field theory, providing a framework to analyze anomaly cancellation using local equivariant cohomology and the variational bicomplex.

Contribution

It introduces a geometric interpretation of locality conditions for anomalies via local equivariant cohomology and relates it to jet bundle cohomology through the variational bicomplex, offering new criteria for anomaly cancellation.

Findings

Established a link between local anomalies and local equivariant cohomology.

Derived necessary and sufficient conditions for anomaly cancellation.

Connected cohomology of jet bundles with variational bicomplex theory.

Abstract

The locality conditions for the vanishing of local anomalies in field theory are shown to admit a geometrical interpretation in terms of local equivariant cohomology, thus providing a method to deal with the problem of locality in the geometrical approaches to the study of local anomalies based on the Atiyah-Singer index theorem. The local cohomology is shown to be related to the cohomology of jet bundles by means of the variational bicomplex theory. Using these results and the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gel'fand-Fuks cohomology introduced in [6], we obtain necessary and sufficient conditions for the cancellation of local gravitational and mixed anomalies.