Covariant Anomalies and Functional Determinants

TL;DR

This paper explores the algebraic structure of gauge and gravitational anomalies, using functional determinants to connect algebraic solutions with effective actions, especially focusing on the Lorentz anomaly and chiral spinors in curved space.

Contribution

It introduces a unified algebraic framework for anomalies and derives explicit functional determinant representations for effective actions in curved backgrounds.

Findings

Unified description of anomalies via complex Lie algebra extension

Functional determinants for covariant anomalies in curved space

Explicit effective action for chiral spinors with U(1) gauge connection

Abstract

We analize the algebraic structure of consistent and covariant anomalies in gauge and gravitational theories: using a complex extension of the Lie algebra it is possible to describe them in a unified way. Then we study their representations by means of functional determinants, showing how the algebraic solution determines the relevant operators for the definition of the effective action. Particular attention is devoted to the Lorentz anomaly: we obtain by functional methods the covariant anomaly for the spin-current and for the energy-momentum tensor in presence of a curved background. With regard to the consistent sector we are able to give a general functional solution only for $d=2$: using the characterization derived from the extended algebra, we find a continuous family of operators whose determinant describes the effective action of chiral spinors in curved space. We compute this action and we generalize the result in presence of a $U(1)$ gauge connection. Accepted for publication in Fortschritte der Physik.