Geodesic diagrams, gravitational interactions & OPE structures

We give a systematic procedure to evaluate conformal partial waves involving symmetric tensors for an arbitrary CFT_d using geodesic Witten diagrams in AdS_d+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.