Holographic symmetry algebra for the MHV sector revisited

TL;DR

This paper revisits the holographic symmetry algebra in the MHV sector, revealing an infinite-dimensional Abelian algebra and its role in deriving missing equations for MHV amplitudes.

Contribution

It identifies the complete symmetry algebra in the MHV sector as a semidirect product involving the $w_{1+ abla}$ algebra and null states, clarifying the algebraic structure and missing equations.

Findings

The symmetry algebra in the MHV graviton sector is a semidirect product of $w_{1+ abla}$ and an Abelian algebra.

In the MHV gluon sector, the algebra is a semidirect product of the $S$ algebra and an Abelian algebra.

Null states in the extended algebra lead to the derivation of two missing KZ-type equations.

Abstract

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.