Classical and holographic aspects of conformal gravity in four dimensions

Abstract

We formulate new boundary conditions that prove well defined variational principle and finite response functions for conformal gravity (CG). In the Anti--de Sitter/conformal field theory framework, gravity theory that is considered in the bulk gives information about the corresponding boundary theory. The metric is split in the holographic coordinate, used to approach the boundary, and the metric at the boundary. One can consider the quantities in the bulk perturbing the (one dimension lower) boundary metric in holographic coordinate. The response functions to fluctuations of the boundary metric are Brown--York stress energy tensor sourced by the leading term in the expansion of the boundary metric and a Partially Massless Response, specific for CG and sourced by the subleading term in the expansion of the boundary metric. They formulate boundary charges that define the asymptotic symmetry algebra or the algebra of the diffeomorphisms that preserve the boundary conditions of the theory. We further analyse CG via canonical analysis constructing the gauge generators of the canonical charges that agree with Noether charges, while the charge associated to Weyl transformations, vanishes. Asymptotic symmetry algebra is determined by the leading term in the expansion of the boundary metric and for the asymptotically Minkowski, R \times S2, and the boundaries related by conformal rescaling, defines conformal algebra. The key role is played by the subleading term in the expansion of the metric forbidden by Einstein gravity equations of motion, however allowed in CG. We classify the subalgebras of conformal algebra restricted by this term and use them to deduce the global solutions of CG. Further, we compute the one loop partition function of CG in four and six dimensions and obtain its structure.