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We formulate new boundary conditions that prove well defined variationalprinciple and finite response functions for conformal gravity (CG). In theAnti--de Sitter/conformal field theory framework, gravity theory that isconsidered in the bulk gives information about the corresponding boundarytheory. The metric is split in the holographic coordinate, used to approach theboundary, and the metric at the boundary. One can consider the quantities inthe bulk perturbing the (one dimension lower) boundary metric in holographiccoordinate. The response functions to fluctuations of the boundary metric areBrown--York stress energy tensor sourced by the leading term in the expansionof the boundary metric and a Partially Massless Response, specific for CG andsourced by the subleading term in the expansion of the boundary metric. Theyformulate boundary charges that define the asymptotic symmetry algebra or thealgebra of the diffeomorphisms that preserve the boundary conditions of thetheory. We further analyse CG via canonical analysis constructing the gaugegenerators of the canonical charges that agree with Noether charges, while thecharge associated to Weyl transformations, vanishes. Asymptotic symmetryalgebra is determined by the leading term in the expansion of the boundarymetric and for the asymptotically Minkowski, R \times S2, and the boundariesrelated by conformal rescaling, defines conformal algebra. The key role isplayed by the subleading term in the expansion of the metric forbidden byEinstein gravity equations of motion, however allowed in CG. We classify thesubalgebras of conformal algebra restricted by this term and use them to deducethe global solutions of CG. Further, we compute the one loop partition functionof CG in four and six dimensions and obtain its structure.

Classical and holographic aspects of conformal gravity in four dimensions