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The Celestial Holographic Dictionary

While our understanding of celestial CFT remains primitive, the intrinsic definition of the CCFT should include a set of rules for constructing consistent (gravitational) scattering amplitudes as a function of their on-shell data. Recent successes demonstrate that the operator product expansion (OPE) data of the CCFT encode the behavior of scattering in collinear limits and can be elegantly determined by global chiral-Poincaré symmetry. This unique feature of CCFT highlights the power of our symmetry-based approach, enabling 4D kinematics to determine the dynamics of the 2D dual. Once we understand the spectrum of operators that capture the bulk scattering states, exploiting the OPE in the 2D CFT would amount to bootstrapping amplitudes from their collinear limits. Therefore, below, we delve into the spectrum from three perspectives: the single-particle operators, the light-ray operators in CCFT, and the multi-particle operators.

Single-particle Operators: Celestial Amplitudes

One key entry in the holographic dictionary is the so-called celestial amplitude, which recast the S-matrix into a basis of boost eigenstates, transforms like the correlation functions of a conformal field theory, and makes manifest infinite dimensional symmetry enhancements that we discussed in the previous section and are typically hidden in soft theorems. The primary operators defined in this "extrapolate"-style manner are single-particle states in CCFTs. For massless particles, celestial amplitudes are related to the momentum space amplitudes via a Mellin transform in the energy. This concrete mapping enables us to identify the celestial avatar of various amplitude properties. For instance, in recent works, we demonstrated the celestial incarnations for dual superconformal symmetry in 4D N=4 super-Yang-Mills [2106.16111] and on-shell recursion relations [2208.11635]. 

Light-ray Operators in Celestial CFTs

In generic CFTs, the OPE coefficients are related to three-point functions and four-point functions contain information about the spectrum of the theory, which can be deduced by means of the conformal block decomposition. In CCFT, (tree-level) 3- and 4-point correlators make these relationships opaque due to their distributional nature resulting from the presence of the momentum-conserving delta function in scattering amplitudes. This implies that CCFT may be an exotic 2D CFT. To harness standard 2D CFT machinery, we could make correlation functions take standard form via integral transformations. It was shown that certain three-point correlators involving light-ray operators take the form of standard three-point CFT correlators. To further study the light-ray operators in CCFT, we computed four-point correlators in (2,2) spacetime signature as well as their conformal block decompositions in [2203.04255] and [2206.08875]. These correlators are non-distributional and allow us to verify that light-ray operators appear in the leading celestial OPEs as expected from a generic Lorentzian CFT perspective. 

Multi-particle Operators

It's worth noting that in the phase space representation, the emergence of two-particle and multi-particle operators is a natural consequence, essential for closing the algebra. This construction implies that multi-particle states should be incorporated into the spectrum of operators within CCFTs. Furthermore, these multi-particle states provide insight into resolving some conundrums inherent to CCFTs, such as the issue of the locality on the complexified celestial sphere and OPE associativity as discussed in [2309.16602]. 

The Celestial Holographic Dictionary: Work
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