Boundary terms of conformal anomaly

TL;DR

The paper develops a unified framework for boundary contributions to the integrated conformal anomaly, showing that boundary terms corresponding to Weyl-invariant bulk invariants are of Hawking–Gibbons type. Using Barvinsky's auxiliary-field method, it derives explicit boundary completions in $d=4$ and $d=6$, and demonstrates agreement with free-field tests in four dimensions. It also points out nonvanishing integrated anomalies in odd dimensions due to boundary terms and provides explicit structures in $d=3$ and $d=5$ for these cases. The work lays groundwork for a systematic classification of conformal boundary invariants and prompts further exploration of the physical meaning of boundary charges and their connection to boundary CFT data and holography.

Abstract

We analyze the structure of the boundary terms in the conformal anomaly integrated over a manifold with boundaries. We suggest that the anomalies of type B, polynomial in the Weyl tensor, are accompanied with the respective boundary terms of the Gibbons-Hawking type. Their form is dictated by the requirement that they produce a variation which compensates the normal derivatives of the metric variation on the boundary in order to have a well-defined variational procedure. This suggestion agrees with recent findings in four dimensions for free fields of various spin. We generalize this consideration to six dimensions and derive explicitly the respective boundary terms. We point out that the integrated conformal anomaly in odd dimensions is non-vanishing due to the boundary terms. These terms are specified in three and five dimensions.