An embedding formalism for CFTs in general states on curved backgrounds

We present a generalisation of the embedding space formalism to conformal field theories (CFTs) on non-trivial 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 CFT${}_d$ lives. We propose a construction of CFT${}_d$ $n$-point functions in non-trivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of non-vanishing 1-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to 2-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.