Invariant Effective Actions, Cohomology of Homogeneous Spaces and Anomalies

TL;DR

This paper systematically constructs the most general local effective actions for Goldstone bosons from symmetry breaking, linking cohomology, anomalies, and topological terms in various spacetime dimensions.

Contribution

It provides an explicit construction of all cohomology generators for any coset space G/H, clarifying their relation to anomalies and topological actions.

Findings

Explicit cohomology generators for G/H coset spaces.

Connection between anomalies and Wess-Zumino-Witten terms.

Construction of gauge actions related to chiral anomalies.

Abstract

We construct the most general local effective actions for Goldstone boson fields associated with spontaneous symmetry breakdown from a group $G$ to a subgroup $H$. In a preceding paper, it was shown that any $G$-invariant term in the action, which results from a non-invariant Lagrangian density, corresponds to a non-trivial generator of the de Rham cohomology classes of $G/H$. Here, we present an explicit construction of all the generators of this cohomology for any coset space $G/H$ and compact, connected group $G$. Generators contributing to actions in 4-dimensional space-time arise either as products of generators of lower degree such as the Goldstone-Wilczek current, or are of the Wess-Zumino-Witten type. The latter arise if and only if $G$ has a non-zero $G$-invariant symmetric $d$-symbol, which vanishes when restricted to the subgroup $H$, i.e. when $G$ has anomalous representations in which $H$ is embedded in an anomaly free way. Coupling of additional gauge fields leads to actions whose gauge variation coincides with the chiral anomaly, which is carried here by Goldstone boson fields at tree level. Generators contributing to actions in 3-dimensional space-time arise as Chern-Simons terms evaluated on connections that are composites of the Goldstone field.