  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.
