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${}_d$ correlators in non-trivial states and on curved
backgrounds using appropriate geometric invariants of the ambient space as
building blocks. As a test of the formalism we apply it to thermal 2-point
functions and find exact agreement with a holographic computation and
expectations based on thermal operator product expansions (OPEs).

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