Recently in their celestial holography programme, Strominger and coworkers attempt to provide a holographic description of conventional 4d gravity. In their investigations, they uncovered a hidden w-infinity symmetry in their `celestial soft OPEs' for graviton scattering. This talk will explain the origin of this symmetry in terms of old ideas of Newman and Penrose based on light-cone cuts of null infinity and their description in terms of asymptotic twistors and certain sigma models in asymptotic twistor space. W_n symmetries were introduced by Zamolodchikov as higher spin symmetries in 2d conformal field theories. These were given a geometric interpretation for n=infinity as area-preserving diffeomorphisms of the plane. I will explain how the corresponding loop algebra becomes a hidden symmetry of self-dual gravity via Penrose's nonlinear graviton construction. The action of this symmetry on the tree-level S-matrix of full gravity beyond the self-dual sector will then be obtained from its action on a sigma model in the asymtotic twistor space of a general space-time. This talk is based on https://arxiv.org/abs/2110.06066 and https://arxiv.org/abs/2103.16984.

Ambitwistor strings are holomorphic string theories whose target space is the space of complex null geoedesics in complexified space-times. I will explain how these theories explain the origin of the scattering equations in twistor strings and the CHY formulae in arbitrary dimensions and provide a reformulation of standard gauge, gravity and other theories in a holomorphic infinite tension analogue of conventional string theories. I will show how these results extend to 1-loop both on a torus and on a nodal Riemann sphere, and perhaps to higher loops.

We use an observation of Maldacena to argue that gravity tree-amplitudes can be computed by inserting appropriate wave-functions into the Witten-Berkovits twistor-string for conformal gravity. This is used to motivate and partially explain the recent remarkable twistor-string-like formula of Cachazo & Skinner that expresses the complete tree-level S-matrix for N=8 supergravity in terms of an integral over rational curves in twistor space.

Leading singularities are invariants of multi-loop scattering amplitudes (the full amplitude at tree level) obtained by generalized unitarity. In this talk we show how to construct multi-loop leading singularities on twistor space for maximally super-symmetric Yang-Mills (and gravity). Building on the tree-level twistor-string representation of scattering amplitudes, they can be represented as integrals over a moduli space of nodal curves in twistor space. We discuss how this might arise from a conjectural twistor-string path integral representation for the full loop amplitude. We also show how the construction relates to the Grassmannian representation of leading singularities conjectured by Arkani-Hamed et. al.. This shows firstly that all leading singularities can be represented in the Grassmannian, and secondly that the complexity is limited, in particular we conjecture that there are no new leading singularities at beyond 3p loops for NpMHV amplitudes.

The twistor programme was introduced by Roger Penrose as an approach to quantum gravity in which twistor space should provide the primary geometric background for physics from which space-time should emerge. This talk will review the programme, i.e., the early successes in formulating the self-dual parts of Yang-Mills and gravity on twistor space. It will go on to review the impact of twistor-string theory, in giving at least a perturbative approach to full Yang Mills and conformal gravity, and outline arguments that prove the equivalence between the twistor-string models and the space-time theories. Finally, twistor-string models for Einstein gravity will be reviewed.