Celestial 𝓌₁₊∞ Symmetries from Twistor Space

Abstract

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 non-linear 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 tree-level 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 I. 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.