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Anatomy of Geodesic Witten Diagrams

Heng-Yu Chen, En-Jui Kuo, Hideki Kyono

TL;DR

The paper clarifies how Geodesic Witten Diagrams provide a constructive holographic realization of conformal blocks, first for scalars and then extending to external spinning operators. By cutting four-point AdS correlators into three-point GWDs and gluing via the spectral parameter, it derives integral representations matching conformal partial waves and shadow structures, and systematically builds spinning blocks using box and differential tensor bases. The work further leverages the split representation to decompose ordinary Witten diagrams with spin-J exchange into sums over geodesic diagrams, clarifying single-trace and double-trace contributions and ensuring consistency through pole-cancelation arguments. Overall, it advances a practical, diagrammatic route to constructing and decomposing spinning conformal blocks within AdS/CFT, with explicit results for key spin configurations and a framework for generalization.

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.

Anatomy of Geodesic Witten Diagrams

TL;DR

The paper clarifies how Geodesic Witten Diagrams provide a constructive holographic realization of conformal blocks, first for scalars and then extending to external spinning operators. By cutting four-point AdS correlators into three-point GWDs and gluing via the spectral parameter, it derives integral representations matching conformal partial waves and shadow structures, and systematically builds spinning blocks using box and differential tensor bases. The work further leverages the split representation to decompose ordinary Witten diagrams with spin-J exchange into sums over geodesic diagrams, clarifying single-trace and double-trace contributions and ensuring consistency through pole-cancelation arguments. Overall, it advances a practical, diagrammatic route to constructing and decomposing spinning conformal blocks within AdS/CFT, with explicit results for key spin configurations and a framework for generalization.

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.

Paper Structure

This paper contains 12 sections, 153 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Scalar Four Point Geodesic Witten Diagrams Revisited
  3. Spinning Three Point Functions and Conformal Blocks
  4. Spinning Conformal Partial Waves from Anti-de Sitter Space
  5. The $(l_1,l_2,0)$ case
  6. The $(1,1,2)$ case
  7. Decomposition of Witten Diagrams via Split Representation
  8. Appendix: Embedding Formalism
  9. Integrals for Three Point Geodesic Witten Diagrams
  10. Integrals for Three Point normal Witten Diagrams
  11. Rewriting tensor structures and some useful identities
  12. Computational Details for Decomposition Analysis

Figures (4)

  • Figure 1: Four point scalar geodesic Witten diagram where the orange curves describe the geodesics and the blue lines are bulk to boundary propagators, such that the interaction vertices move along geodesics $\gamma_{12}$ and $\gamma_{34}$.
  • Figure 2: Cutting the four point scalar geodesic Witten diagram into the three point ones.
  • Figure 3: Three point scalar geodesic Witten diagram
  • Figure 4: Left: Normal exchange Witten diagram with four scalar external fields and a spin-$J$ internal field, Right: A split diagram