MHV Graviton Scattering Amplitudes and Current Algebra on the Celestial Sphere

TL;DR

The paper identifies an autonomous MHV sector of the Celestial CFT, governed by an extended symmetry algebra consisting of a level-zero overline{SL(2,C)} current algebra and supertranslations from the positive-helicity soft graviton. By deriving the subleading and leading soft theorems as Ward identities, it constructs the corresponding currents and their commutators with SL(2,C) and among themselves, showing a closed algebra that constrains the OPEs of graviton primaries. Using Hodges' formula for tree-level MHV amplitudes and their Mellin transforms, the authors derive explicit leading and subleading celestial OPEs and demonstrate that null states of the extended algebra enforce first-order PDEs on MHV amplitudes. These structures imply an autonomous, solvable sector of the Celestial CFT that holographically encodes MHV graviton scattering, with differential equations and beta-function-type coefficients tying OPE data directly to symmetry requirements. The work also provides detailed appendix-derived checks and recursive relations for subleading coefficients, laying groundwork for a full algebraic solution of the MHV sector in celestial holography.

Abstract

The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for $\overline{SL(2,\mathbb C)}$ currents. This naturally gives rise to a $\overline{SL(2,\mathbb C)}$ current algebra living on the celestial sphere. The generators of the $\overline{SL(2,\mathbb C)}$ 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.