Spacetime $Lw_{1+\infty}$ Symmetry and Self-Dual Gravity in Plebanski Gauge

The space of self-dual Einstein spacetimes in 4 dimensions is acted on by an infinite dimensional Lie algebra called the $Lw_{1+\infty}$ algebra. In this work we explain how one can ``build up'' self-dual metrics by acting on the flat metric with an arbitrary number of infinitesimal $Lw_{1+\infty}$ transformations, using a convenient choice of gauge called Plebanski gauge. We accomplish this through the use of something called a ``perturbiner expansion,'' which will perturbatively generate for us a self-dual metric starting from an initial set of quasinormal modes called integer modes. Each integer mode corresponds to a particular $Lw_{1+\infty}$ transformation, and this perturbiner expansion of integer modes will be written as a sum over ``marked tree graphs,'' instead of momentum space Feynman diagrams. We find that a subset of the $Lw_{1+\infty}$ transformations act as spacetime diffeomorphisms, and the algebra of these diffeomorphisms is $w_{\infty} \ltimes f$. We also show all analogous results hold for the $Ls$ algebra in self-dual Yang Mills.