Holographic symmetry algebra for the MHV sector revisited

We revisit the holographic symmetry algebra in the MHV sector. We find an infinite dimensional Abelian symmetry algebra whose generators are the conformally soft negative helicity gravitons and gluons. So the complete symmetry algebra in the MHV graviton sector is a semideirect product of the $w_{1+\infty}$ algebra and the infinite dimensional Abelian algebra. Similarly in the MHV gluon sector the symmetry algebra is a semidirect product of the $S$ algebra and the infinite dimensional Abelian algebra. The extended symmetry algebra has some use. For example, it is known for sometime that an $n$ point MHV amplitude satisfies $(n-2)$ Knizhnik-Zamolodchikov (KZ) type equations. So two equations are missing. We show that the extended symmetry algebra has additional null states whose decoupling give rise to the two missing equations.