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.
Highlights
JHEP03(2018)072 as the η(0) invariant of the Dirac operator
We compute the anomaly on 4D manifolds with boundaries with local bag boundary conditions
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
Summary
We shall express the parity anomaly through the spectral η function and compute the variation of the latter in terms of the heat kernel expansion. This method was used in [4, 21, 22]. The paper by Ojima computed a fraction of two determinants, det(D/ − im)/ det(D/ σ=0 − im), see [29, eq (3.3)], where D/ σ=0 means the Dirac operator in curved space but without the spin-connection term. Our results differ from that of [29]
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