Conformal Cauchy Slice Holography: An Alternative Phase Space For Gravity

The phase space of gravitational theories in asymptotically Anti-de Sitter (AAdS) spacetimes consists of geometries, matter configurations, and their conjugate momenta on a Cauchy surface, subject to the Hamiltonian, momentum, and matter-gauge constraints. When a unique maximal volume slice exists in all classical solutions of the bulk equations of motion, and the matter fields satisfy certain conditions, we show that this phase space is physically equivalent to an alternative phase space in which the Hamiltonian constraint is replaced by the real Weyl-anomaly constraint, while the momentum and matter-gauge constraints remain unchanged. A necessary requirement for a functional of the metric and matter configurations to qualify as a valid quantum gravity state is that it satisfies the operator gauge constraints. Partition functions of certain conformal field theories with imaginary central charge, defined on bulk Cauchy slices, satisfy these operator gauge constraints and therefore provide candidate quantum gravity states in the alternative phase space formulation.