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Celestial \(w_{1+\infty}\) symmetries from twistor space. (English) Zbl 1489.83067

Summary: 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.

MSC:

83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
81U20 \(S\)-matrix theory, etc. in quantum theory
32L25 Twistor theory, double fibrations (complex-analytic aspects)
22E67 Loop groups and related constructions, group-theoretic treatment
31C45 Other generalizations (nonlinear potential theory, etc.)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
17B69 Vertex operators; vertex operator algebras and related structures
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