Counterterms and dual holographic anomalies in CS gravity

The holographic Weyl anomaly associated to Chern-Simons gravity in 2n + 1 dimensions is proportional to the Euler term in 2n dimensions, with no contributions from the Weyl tensor. We compute the holographic energy-momentum tensor associated to ChernSimons gravity directly from the action, in an arbitrary odd-dimensional spacetime. We show, in particular, that the counterterms rendering the action finite contain only terms of the Lovelock type. The AdS/CFT correspondence[1, 2, 3] provides a prescription to compute vacuum expectation values of CFT operators in terms of dual classical fields on AdS. This prescription has been checked successfully in many examples. In particular, the CFT energy momentum tensor T ij is dual to the spacetime metric governed by Einstein’s equations with a negative cosmological constant. One can use the correspondence to compute the CFT Weyl anomaly. This calculation was indicated in [3], and carried on in detail in [4] with the expected result 〈T μ μ 〉 = A = N 32π2 ( RRij − 1 3 R ) , (1) for N = 4 SU(N) SYM theory in four dimensions which is dual to type IIB string theory on AdS5 × S. For a CFT (in d = 4) with ns real scalars, nf Dirac spinors and nv vectors the anomaly is [5] A = c C − aE4 (2) with a = 1 360(4π)2 (ns + 11nf + 62nv) (3)