Lie, Noether, Kosmann, and Diffeomorphism Anomalies Redux

TL;DR

This paper clarifies the relationship between Noether, energy-momentum tensors, and anomalies in gauge theories, revealing that the Kosmann lift plays a crucial role in ensuring symmetry and correcting anomaly calculations.

Contribution

It demonstrates that the Noether energy-momentum tensor equals the symmetric tensor using the Lie derivative and Kosmann lift, impacting anomaly computations and clarifying longstanding ambiguities.

Findings

The Noether and symmetric energy-momentum tensors are equal for general tensors.

For spinors, the Kosmann lift ensures the tensor equality and symmetry.

Diffeomorphism anomalies are halved when using the Kosmann lift, without affecting anomaly polynomials.

Abstract

The Noether procedure carries an inherent ambiguity due to the necessary local extension, no longer a symmetry, of the global symmetry. The gauging should fix the ambiguity once and for all, however, and, for translations, the general covariance demands us to use the Lie derivative. We argue that, with this alone and without any further tweaking, the Noether energy-momentum $\hat{\mathbb{T}}$ must equal the symmetric counterpart, $T$, inevitably and show the equality explicitly for general tensors. For spinors, a subtlety with the Lie derivative itself enters the issue and leads us to the Kosmann lift, often unnoticed by the physics community, from which $T=\hat{\mathbb{T}}$ again emerges straightforwardly and in a naturally symmetric form. Finally, we address how the same Kosmann lift affects the anomaly computations and show that the diffeomorphism anomaly from the seminal paper must be halved, while the venerable anomaly polynomials themselves stand unaffected. We discuss the ramifications of these findings.