Holographic Description of Gravitational Anomalies

Abstract

The holographic duality can be extended to include quantum theories with broken coordinate invariance leading to the appearance of the gravitational anomalies. On the gravity side one adds the gravitational Chern-Simons term to the bulk action which gauge invariance is only up to the boundary terms. We analyze in detail how the gravitational anomalies originate from the modified Einstein equations in the bulk. As a side observation we find that the gravitational Chern-Simons functional has interesting conformal properties. It is invariant under conformal transformations. Moreover, its metric variation produces conformal tensor which is a generalization of the Cotton tensor to dimension $d+1=4k-1, k\in Z$. We calculate the modification of the holographic stress-energy tensor that is due to the Chern-Simons term and use the bulk Einstein equations to find its divergence and thus reproduce the gravitational anomaly. Explicit calculation of the anomaly is carried out in dimensions $d=2$ and $d=6$. The result of the holographic calculation is compared with that of the descent method and agreement is found. The gravitational Chern-Simons term originates by Kaluza-Klein mechanism from a one-loop modification of M-theory action. This modification is discussed in the context of the gravitational anomaly in six-dimensional $(2,0)$ theory. The agreement with earlier conjectured anomaly is found.