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

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