In this note we determine the graviton-graviton OPE and the null states in any w 1+∞ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete infinite family of such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite family. However, the bulk description of other theories are not known to us.

In this note we determine the graviton-graviton OPE and the null states in any w1+∞ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete 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.