Gravitational parity anomaly with and without boundaries

TL;DR

This paper investigates the gravitational parity anomaly in three and four dimensions, focusing on boundary effects, localizes the anomaly on boundaries, and explores its properties and relations across dimensions.

Contribution

It provides a re-computation of the gravitational parity anomaly in 3D and extends analysis to 4D manifolds with boundaries, revealing boundary-localized anomalies with Chern-Simons and extrinsic curvature terms.

Findings

Anomaly localized on the boundary in 4D with boundary conditions

Presence of gravitational Chern-Simons terms in the boundary anomaly

Discussion of conformal invariance and dimensional relations

Abstract

In this paper we consider gravitational parity anomaly in three and four dimensions. We start with a re-computation of this anomaly on a 3D manifold without boundaries and with a critical comparison of our results to the previous calculations. Then we compute the anomaly on 4D manifolds with boundaries with local bag boundary conditions. We find, that gravitational parity anomaly is localized on the boundary and contains a gravitational Chern-Simons terms together with a term depending of the extrinsic curvature. We also discuss the main properties of the anomaly, as the conformal invariance, relations between 3D and 4D anomalies, etc.