# Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

## TL;DR

This paper develops a systematic holographic representation for higher-point conformal blocks in AdS/CFT, extending known four-point results to five and six points using new propagator identities, and introduces techniques for decomposing higher-point AdS diagrams.

## Contribution

It introduces higher-point propagator identities and geodesic diagram techniques to compute higher-point conformal blocks and AdS diagrams, extending previous four-point results.

## Key findings

- Explicit holographic representations for five- and six-point conformal blocks.
- Closed-form expressions for higher-point AdS diagram decomposition coefficients.
- A new algebraic understanding of singularities in higher-point AdS diagrams.

## Abstract

Conformal blocks are the fundamental, theory-independent building blocks in any CFT, so it is important to understand their holographic representation in the context of AdS/CFT. We describe how to systematically extract the holographic objects which compute higher-point global (scalar) conformal blocks in arbitrary spacetime dimensions, extending the result for the four-point block, known in the literature as a geodesic Witten diagram, to five- and six-point blocks. The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime. Using the holographic representation of the higher-point conformal blocks and higher-point propagator identities, we develop geodesic diagram techniques to obtain the explicit direct-channel conformal block decomposition of a broad class of higher-point AdS diagrams in a scalar effective bulk theory, with closed-form expressions for the decomposition coefficients. These methods require only certain elementary manipulations and no bulk integration, and furthermore provide quite trivially a simple algebraic origin of the logarithmic singularities of higher-point tree-level AdS diagrams. We also provide a more compact repackaging in terms of the spectral decomposition of the same diagrams, as well as an independent discussion on the closely related but computationally simpler framework over $p$-adics which admits comparable statements for all previously mentioned results.