Abstract
We give a systematic procedure to evaluate conformal partial waves involving symmetric tensors for an arbitrary CFTd using geodesic Witten diagrams in AdSd+1. Using this procedure we discuss how to draw a line between the tensor structures in the CFT and cubic interactions in AdS. We contrast this map to known results using three-point Witten diagrams: the maps obtained via volume versus geodesic integrals differ. Despite these differences, we show how to decompose four-point exchange Witten diagrams in terms of geodesic diagrams, and we discuss the product expansion of local bulk fields in AdS.
Highlights
We show how to decompose four-point exchange Witten diagrams in terms of geodesic diagrams, and we discuss the product expansion of local bulk fields in AdS
Our aim here is to apply the efficiency of the conformal block decomposition to holography: can we organize observables in AdS gravity as we do in a CFT? This question has been at the heart of holography since its conception [7,8,9], with perhaps the most influential result the prescription to evaluate CFT correlation functions via Witten diagrams [9]
Even though there are non-trivial cancellations in the geodesic diagrams, in section 5 we show how to decompose four point exchange Witten diagrams in terms of geodesic diagrams
Summary
The simplest way to carry out our analysis is via the embedding space formalism. We will use this to describe both CFTd and AdSd+1 quantities. This formalism was recently revisited and exploited in [33, 42, 43], and we mainly follow their presentation. This section summarises the most important definitions and relations we will use throughout; readers familiar with this material can skip this section. All of our discussion will be in Euclidean signature
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