On the eta-invariant in the 4D chiral U(1) theory

TL;DR

This paper investigates the imaginary part of the effective action in 4D chiral U(1) gauge theory, revealing it generally non-zero but small after anomaly cancellation, with implications for understanding gauge anomalies and eta-invariants.

Contribution

It provides a continuum fermion approach analysis of the eta-invariant's role in the imaginary part of the effective action in 4D chiral U(1) theories, highlighting the effects of anomaly cancellation.

Findings

Perturbative analysis shows a generally non-zero imaginary part.

After anomaly cancellation, the imaginary part is small.

The eta-invariant contributes to the effective action's imaginary component.

Abstract

In contrast to its counterpart in a vector theory, the effective action of a chiral gauge theory may have a non-vanishing imaginary part. It consists of the so-called Chern-Simons form, which encodes the anomaly and a gauge invariant piece related to the eta-invariant of some extended Dirac operator. We investigate the imaginary part of the effective action in the 4D chiral U(1) theory within the continuum fermion approach. Perturbative analysis indicates in general a non-vanishing imaginary part of the effective action. However, after anomaly cancellation we find only a small value of the imaginary part for the investigated configurations.