An infinite family of $w_{1+\infty}$ invariant theories on the celestial sphere

TL;DR

This paper identifies an infinite family of theories on the celestial sphere with $w_{1+ abla}$ symmetry, characterized by constrained graviton scattering amplitudes and null states, including known models like MHV-sector and quantum self dual gravity.

Contribution

It determines the graviton-graviton OPE and null states for all $w_{1+ abla}$ symmetric theories, revealing an infinite family with heavily constrained amplitudes.

Findings

Existence of an infinite family of $w_{1+ abla}$ invariant theories.

Identification of null states constraining graviton scattering amplitudes.

Examples include MHV-sector and quantum self dual gravity.

Abstract

In this note we determine the graviton-graviton OPE and the null states in any $w_{1+\infty}$ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete \textit{infinite} family of such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite family. Although the Bulk Lagrangian description of this family of theories is not currently known to us, the graviton scattering amplitudes in these theories are heavily constrained due to the existence of null states. Presumably they are exactly solvable in the same way as the minimal models of $2$-D CFT.