Celestial w(1+infinity) Symmetries from Twistor Space

We explain how twistor theory represents the self-dual sector of four dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's nonlinear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra Lw1+,,,, of the algebra w1+,,,, of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in treelevel perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector which describes maps from the Riemann sphere to the asymptotic twistor space defined from characteristic data at null infinity 5. We show that the OPE of the sigma model naturally encodes the Poisson structure on twistor space and gives rise to the celestial realization of Lw1+,,,,. The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of w1+,,,, and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of Lw1+,,,,, one for each of the self-dual and anti-self-dual sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor string.