Non-Abelian Anomalies and Effective Actions for a Homogeneous Space $G/H$

TL;DR

This paper develops a method to construct gauge-invariant effective actions for Nambu-Goldstone bosons on homogeneous spaces, addressing anomalies and providing explicit formulas, especially for symmetric spaces.

Contribution

It introduces a general framework for constructing anomaly-compatible effective actions on homogeneous spaces, including explicit counterterms and decompositions for symmetric spaces.

Findings

Effective actions can be constructed when n(G/H) is trivial.

Explicit forms of anomalies and counterterms are provided.

Decomposition into parity even and odd parts is achieved for symmetric spaces.

Abstract

We consider the problem of constructing the fully gauged effective action in $2n$-dimensional space-time for Nambu-Goldstone bosons valued in a homogeneous space $G/H$, with the requirement that the action be a solution of the anomalous Ward identity and be invariant under the gauge transformations of $H$. We show that this can be done whenever the homotopy group $\pi_{2n}(G/H)$ is trivial, $G/H$ is reductive and $H$ is embedded in $G$ so as to be anomaly free, in particular if $H$ is an anomaly safe group. We construct the necessary generalization of the Bardeen counterterm and give explicit forms for the anomaly and the effective action. When $G/H$ is a symmetric space the counterterm and the anomaly decompose into a parity even and a parity odd part. In this case, for the parity even part of the action, one does not need the anomaly free embedding of $H$.