Abstract
The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} currents. This naturally gives rise to a overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} current algebra living on the celestial sphere. The generators of the overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this algebra. To be more precise, 1) The subleading terms in the OPE are determined in terms of the leading OPE coefficient if we demand that both sides of the OPE transform in the same way under this local symmetry algebra. 2) Positive helicity gravitons have null states under this local algebra whose decoupling leads to differential equations for MHV amplitudes. An n point MHV amplitude satisfies two systems of (n − 2) linear first order PDEs corresponding to (n − 2) positive helicity gravitons. We have checked, using Hodges’ formula, that one system of differential equations is satisfied by any MHV amplitude, whereas the other system has been checked up to six graviton MHV amplitude. 3) One can determine the leading OPE coefficients from these differential equations.This points to the existence of an autonomous sector of the Celestial CFT which holographically computes the MHV graviton scattering amplitudes and is completely defined by this local symmetry algebra. The MHV-sector of the Celestial CFT is like a minimal model of 2-D CFT.
Highlights
We find that the operator product expansion (OPE) of two graviton primaries in the Celestial conformal field theories (CFT), extracted from MHV amplitudes, is completely determined in terms of this algebra
S-matrix elements, which are the primary observables of the bulk theory, can be expressed in a particular basis [26, 27] where they manifestly transform as correlation functions of primary operators in 2-D CFT
In this paper our objective is to explore the implications of these symmetries for the operator product expansion (OPE) in Celestial CFTs
Summary
The holographic dual of quantum theories of gravity in four dimensional (4-D) asymptotically flat spacetimes has been conjectured to be two dimensional (2-D) conformal field theories (CFT) which live on the celestial sphere at null infinity [9,10,11,12,13, 17, 18, 26,27,28,29,30,31,32,33,34,35,36,37,38,39, 50, 51]. This is analogous to null state relations familiar from 2-D CFTs and leads to a first order linear partial differential equation that must be satisfied by the 3-point amplitude. We find that this OPE can be organised according to representations of the extended symmetry algebra that we referred to above.
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