Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 016, 23 pages      arXiv:2110.06066
Contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger Penrose

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

Tim Adamo a, Lionel Mason b and Atul Sharma b
a) School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, EH9 3FD, UK
b) The Mathematical Institute, University of Oxford, OX2 6GG, UK

Received November 22, 2021, in final form February 17, 2022; Published online March 08, 2022

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 $Lw_{1+\infty}$ of the algebra $w_{1+\infty}$ of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton 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 ${\mathcal I}$. 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 $Lw_{1+\infty}$. The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of $w_{1+\infty}$ and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of $Lw_{1+\infty}$, 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.

Key words: twistor theory; scattering amplitudes; self-duality.

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