# Anatomy of Geodesic Witten Diagrams

## TL;DR

This paper provides a detailed analysis of Geodesic Witten Diagrams, establishing their connection to conformal partial waves and extending the framework to include spinning operators and propagators.

## Contribution

It offers a systematic construction of GWDs from CFT operator expansions, including their gluing procedure and generalization to spinning cases, advancing holographic duality understanding.

## Key findings

- Explicit link between GWDs and conformal partial waves
- Decomposition of spinning Witten diagrams into scalar GWDs
- Generalization to spinning operators and propagators

## Abstract

We revisit the so-called "Geodesic Witten Diagrams" (GWDs) \cite{ScalarGWD}, proposed to be the holographic dual configuration of scalar conformal partial waves, from the perspectives of CFT operator product expansions. To this end, we explicitly consider three point GWDs which are natural building blocks of all possible four point GWDs, discuss their gluing procedure through integration over spectral parameter, and this leads us to a direct identification with the integral representation of CFT conformal partial waves. As a main application of this general construction, we consider the holographic dual of the conformal partial waves for external primary operators with spins. Moreover, we consider the closely related "split representation" for the bulk to bulk spinning propagator, to demonstrate how ordinary scalar Witten diagram with arbitrary spin exchange, can be systematically decomposed into scalar GWDs. We also discuss how to generalize to spinning cases.