A Simple Algebraic Derivation of the Covariant Anomaly and Schwinger Term

TL;DR

This paper presents a straightforward algebraic derivation of the covariant anomaly and Schwinger term, clarifying their relationship and differences, especially regarding their local forms in gauge theories.

Contribution

It introduces a simple algebraic expression for the curvature of the covariant determinant line bundle, enabling a unified derivation of anomalies and Schwinger terms.

Findings

Derived covariant anomaly and Schwinger term expressions

Clarified the relationship between consistent and covariant forms

Showed the non-locality of the Schwinger term difference

Abstract

An expression for the curvature of the "covariant" determinant line bundle is given in even dimensional space-time. The usefulness is guaranteed by its prediction of the covariant anomaly and Schwinger term. It allows a parallel derivation of the consistent anomaly and Schwinger term, and their covariant counterparts, which clarifies the similarities and differences between them. In particular, it becomes clear that in contrary to the case for anomalies, the difference between the consistent and covariant Schwinger term can not be extended to a local form on the space of gauge potentials.