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We explain how twistor theory represents the self-dual sector of fourdimensional gravity in terms of the loop group of Poisson diffeomorphisms ofthe plane via Penrose's non-linear graviton construction. The symmetries of theself-dual sector are generated by the corresponding loop algebra$Lw_{1+\infty}$ of the algebra $w_{1+\infty}$ of these Poisson diffeomorphisms.We show that these coincide with the infinite tower of soft graviton symmetriesin tree-level perturbative gravity recently discovered in the context ofcelestial amplitudes. We use a twistor sigma model for the self-dual sectorwhich describes maps from the Riemann sphere to the asymptotic twistor spacedefined from characteristic data at null infinity ${\mathcal I}$. We show thatthe OPE of the sigma model naturally encodes the Poisson structure on twistorspace and gives rise to the celestial realization of $Lw_{1+\infty}$. Thevertex operators representing soft gravitons in our model act as currentsgenerating the wedge algebra of $w_{1+\infty}$ and produce the expectedcelestial OPE with hard gravitons of both helicities. We also discuss how thetwo copies of $Lw_{1+\infty}$, one for each of the self-dual and anti-self-dualsectors, are represented in the OPEs of vertex operators of the 4d ambitwistorstring.

Celestial $w_{1+\infty}$ Symmetries from Twistor Space