Embedding formalism for CFTs in general states on curved backgrounds

Abstract

We present a generalization of the embedding space formalism to conformal field theories (CFTs) on nontrivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in $d+2$ dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a $d$-dimensional conformal manifold, which is the physical spacetime where the ${\mathrm{CFT}}_{d}$ lives. We propose a construction of ${\mathrm{CFT}}_{d}$ $n$-point functions in nontrivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of nonvanishing one-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to two-point functions of thermal CFT, finding exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs), and to CFTs on squashed spheres where no prior results are known and existing methods are difficult to apply, demonstrating the utility of the method.