# Geodesic Diagrams, Gravitational Interactions & OPE Structures

## TL;DR

This paper develops a systematic method to evaluate conformal partial waves with symmetric tensors in any CFT$_d$ using geodesic Witten diagrams in AdS$_{d+1}$, linking CFT tensor structures to AdS cubic interactions.

## Contribution

It introduces a new procedure for computing conformal partial waves with tensors and clarifies the relationship between CFT tensor structures and AdS cubic interactions.

## Key findings

- Devised a systematic evaluation method for conformal partial waves with tensors.
- Established a connection between CFT tensor structures and AdS cubic interactions.
- Demonstrated decomposition of four-point exchange Witten diagrams into geodesic diagrams.

## Abstract

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.