Gravity from holomorphic discs and celestial $Lw_{1+\infty}$ symmetries

In split or Kleinian signature, twistor constructions parametrize solutions to both gauge and gravity self-duality (SD) equation from twistor data that can be expressed in terms of free smooth data without gauge freedom. Here the corresponding constructions are given for asymptotically flat SD gravity providing a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function $h$ on the real twistor space. Geometrically $h$ determines a nonlinear deformation of the location of the real twistor space $\RP^3$ inside the complex twistor space $\CP^3$. This presentation gives an optimal presentation of Strominger's recently discovered $Lw_{1+\infty}$ celestial symmetries. These, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. However, when imaginary, such Poisson transformations are active symmetries, and generate changes to the gravitational field by deforming the location of the real slice of the twistor space. Gravity amplitudes for the full, non-self-dual Einstein gravity, arise as correlators of a chiral twistor sigma model. This is reformulated for split signature as a theory of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by $h$. Real $Lw_{1+\infty}$ symmetries act oas gauge symmetries, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, an analogous framework for Yang-Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.