Equating Extrapolate Dictionaries for Massless Scattering

TL;DR

This paper explores how conformal inversions relate different celestial CFT correlators and boost eigenstates, providing new insights into the extrapolate dictionary and the flat space limit of AdS/CFT.

Contribution

It demonstrates the role of conformal inversions in connecting boost eigenstates to shadow states and unifies various extrapolate dictionaries within celestial CFT.

Findings

Conformal inversions map boost eigenstates to shadow states.

Boundary correlators match celestial amplitudes of shadow transformed states.

Provides a unified framework for extrapolate dictionaries and flat limit of AdS/CFT.

Abstract

We study features of celestial CFT correlation functions when the bulk theory is itself a CFT. We show that conformal inversions in the bulk map boost eigenstates to shadow transformed boost eigenstates. This is demonstrated explicitly for the wavefunctions of free massless scalars, and finds interesting applications to building extrapolate dictionaries. Because inversions exchange null infinity and the light cone of the origin, one finds a relation between the massless extrapolate dictionary -- involving correlators of operators inserted along null infinity -- and the slice-by-slice extrapolate dictionary recently studied by Sleight and Taronna starting from the hyperbolic foliation of de Boer and Solodukhin. Namely, boundary correlators of Sleight and Taronna coincide with celestial amplitudes of shadow transformed boost eigenstates. These considerations are unified by lifting celestial correlators to the Einstein cylinder. This also sheds new light on the extraction of the $S$-matrix from the flat limit of AdS/CFT.