Holographic dual of the five-point conformal block

TL;DR

The paper constructs a holographic object for the five-point global conformal block in $AdS_{n+1}/CFT_n$ as a weighted sum of five-point geodesic bulk diagrams, extending the four-point geodesic framework. It proves the construction reproduces the five-point block by showing the sum is an eigenfunction of the conformal Casimir with the appropriate boundary conditions, with guidance from a $p$-adic Bruhat--Tits tree. A detailed OPE-limit check and embedding-space Casimir analysis underpin the Casimir-based verification, and a nontrivial propagator identity inspired by the $p$-adic case is pivotal. The work opens avenues for higher-point blocks, spins, and loop computations, and aligns with known explicit five-point blocks while offering a constructive holographic perspective.

Abstract

We present the holographic object which computes the five-point global conformal block in arbitrary dimensions for external and exchanged scalar operators. This object is interpreted as a weighted sum over infinitely many five-point geodesic bulk diagrams. These five-point geodesic bulk diagrams provide a generalization of their previously studied four-point counterparts. We prove our claim by showing that the aforementioned sum over geodesic bulk diagrams is the appropriate eigenfunction of the conformal Casimir operator with the right boundary conditions. This result rests on crucial inspiration from a much simpler $p$-adic version of the problem set up on the Bruhat-Tits tree.

Figures (4)

• Figure 1: Obtaining the conformal block decomposition of the four-point exchange diagram Hijano:2015zsa. Green, disk-shaped bulk vertices are to be integrated over all of AdS. The remaining bulk vertices are integrated over boundary anchored geodesics, shown as dashed, red curves. Just for this figure, we have used the shorthand: $\Delta_{k_1} = \Delta_1+\Delta_2+2k_1, \Delta_{k_2} = \Delta_3+\Delta_4+2k_2$. We used (\ref{['KKexpansion']}) to get to the second step, and (\ref{['GG']}) to obtain the third step. The ellipsis in the third step corresponds to contributions from double-trace exchanges.
• Figure 2: The geodesic integral of (\ref{['4ptGeodesic']}), represented diagrammatically as a geodesic bulk diagram. The dashed, red curves represent the boundary anchored geodesics along which the respective bulk interaction vertices are integrated. In the literature, this is often referred to as a (four-point) geodesic Witten diagram Hijano:2015zsa. In particular, the same diagram computes the $p$-adic four-point conformal partial wave when evaluated on the Bruhat--Tits tree instead of AdS Gubser:2017tsi. (In later figures, locations of operator insertions will often be suppressed when no confusion is likely.)
• Figure 3: Attempting a conformal block decomposition of a five-point bulk diagram. Just for this figure, we have used the shorthand: $\Delta_{k_1} = \Delta_1+\Delta_2+2k_1, \Delta_{k_2} = \Delta_4+\Delta_5+2k_2$. The figure in the third step shows the unevaluated $\int \hat{K}\hat{G}\hat{G}$ AdS integral.
• Figure 4: The five-point geodesic bulk diagram on the Bruhat--Tits tree dual to the five-point $p$-adic conformal partial wave JP:inprep.