Gravity and Lw1+∞ Symmetry

Abstract

We develop a new perturbative expansion for self-dual metrics in four-dimensions using Plebański's second heavenly equation. The expansion can be expressed as a sum over "marked" tree graphs, where each node can have an arbitrary number of edges. The expansion can equivalently be expressed as a sum over binary tree graphs. We then use the expansion for two different applications. In the first application, we provide a direct derivation of the NSVW tree-formula for the graviton MHV amplitude by evaluating the gravitational action on-shell in the saddle-point approximation. In the second application, we use the perturbiner expansion to write down an explicit equation for the action of the Lw_{1+∞} algebra on the spacetime metric. To be more precise, we give a formula for the metric after it has been acted on by N arbitrary Lw_{1+∞} transformations, for any finite N, assuming the metric started as the flat metric. This formula holds everywhere in the bulk. Each Lw_{1+∞} transformation can be understood as adding a positive helicity graviton into the spacetime, where the wavefunction of said graviton takes a very specific form. The order in which these gravitons are added matters, and the commutation relation of this infinitesimal action is precisely that of the Lw_{1+∞} algebra.