# RP90: Twistors from Geometry to Physics

This zoom meeting will mark the 90th birthday of Roger Penrose with talks on twistor theory, geometry and mathematical physics July 22-23, 2021.

**Email: **@email

**Schedule (UK timings, BST=GMT+1):**

Thursday 22 July:

Chair: Oscar Garcia Prada: Video of session

11am: Nigel Hitchin, *Spinors: twistors and classical geometry*. Slides

12-12.30: Hadleigh Frost, *Lightcones, causality, and Teichmuller spaces*.

12.30-2: Lunch.

Chair: Paul Tod Video of Session

2-3pm: Maciej Dunajski,* Null Kahler geometry, and Twistor Theory*. Slides

3-4pm: David Skinner, *Twistors, Integrability and 4d Chern-Simons Theory*.

4-4.30: break

Chair Anastasia Volovich: Video of session

4.30-5.30: Nima Arkani-Hamed,* Non-perturbative positive geometries in twistor space: from the amplituhedron to AdS.*

Friday 23rd July:

Chair: Tsou Sheung Tsun; Video of session.

11am: Mike Eastwood, *Spinors and five-dimensional contact geometry*.

12: Yvonne Geyer, *From triality in 6d to loops in 4d via ambitwistor strings. *Slides

1-2.30 Lunch break

Chair: Lionel Mason; video of session.

2.30: Claude LeBrun, *Twistors, Self-Duality, and Spin^c Structures. *Slides-corrected-20/11/21

3.30: Tim Adamo, * From non-linear gravitons to gravitational scattering. *Slides

4.30: Tea.

Chair: Malcolm MacCallum; Video of session.

5pm: Roger Penrose, *Twistors, Pseudo-Twistors, and Split-Octonions.*

Talks are expected to be around 50 minutes to allow time for questions and changeover. It is planned to record all talks and to post them to this website in due course.

**Speakers, talk titles and abstracts: **

Tim Adamo (Edinburgh): From non-linear gravitons to gravitational scattering

Abstract: Penrose's non-linear graviton construction is one of the highlights of twistor theory, establishing a one-to-one correspondence between self-dual four-manifolds and twistor spaces. In recent years, this beautiful piece of geometry has shed new light on the computation of gravitational scattering amplitudes. I will discuss two examples of this: firstly, the non-linear graviton provides a derivation of remarkable formulae for tree-level graviton scattering in Minkowski space directly from classical general relativity. But it also allows us to go beyond the current reach of other approaches to perturbative gravity by computing graviton scattering amplitudes in any self-dual radiative space-time.

Nima Arkani-Hamed (IAS, Princeton): Non-perturbative positive geometries in twistor space: from the amplituhedron to AdS

Maciej Dunajski (Cambridge): Null Kahler geometry, and Twistor Theory.

Abstract: We construct the normal forms of null-Kahler metrics: split signature metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear in the Bridgeland stability conditions of the moduli spaces of Calabi-Yau three-folds.

In dimension 4, the split signature version of Penrose's Nonlinear Graviton theorem gives a connection with integrable systems and Painleve equations.

Mike Eastwood (Adelaide): Spinors and five-dimensional contact geometry

Abstract: Spinors and space-time, CUP 1984 and 1986, are well-known landmarks in general relativity and twistor theory. Spinors are essential in formulating the Dirac equation and Roger Penrose's twistor equation. But it does not stop there! Spinors can also be used to define various geometric structures in five dimensions. There are some remarkable consequences, one of which is implicit in Engel's 1893 construction of the exceptional Lie algebra G2. This talk is based on joint work with Pawel Nurowski and Timothy Moy.

Hadleigh Frost (Oxford): Lightcones, causality, and Teichmuller spaces

Yvonne Geyer (Chulalongkorn): From triality in 6d to loops in 4d via ambitwistor strings

Abstract: Ambitwistor strings — chiral worldsheet models that map into ambitwistor space — are generalizations of Witten's twistor string whose correlators naturally lead to compact amplitude representations for massless field theories. In this talk, I will discuss a string built on the twistor representation of the ambitwistor space in 6d, describing (conformally invariant) biadjoint scalar theory. Reflecting triality for the conformal group, three such models can be formulated. On reduction to 5d, we find two additional worldsheet models, describing super Yang-Mills theory and gravity. Correlators in these models lead to amplitude formulae that are manifestly supersymmetric. I conclude by discussing a further symmetry reduction to 4d, which allows us to describe massive particles, as well as one-loop amplitudes obtained via a so-called 'gluing operator'.

Nigel Hitchin (Oxford): Spinors: twistors and classical geometry

Abstract: Since the 1970s when I first learned about twistor spaces, linear and nonlinear, they have always held an attraction for me. The talk will focus on the way they have appeared in some recent investigations into concrete examples of gauge-theoretic moduli spaces. The examples lead back to classical constructions in projective geometry which have appealed to mathematicians over the ages.

Claude LeBrun (Stonybrook): Twistors, Self-Duality, and Spin^c Structures

Abstract: Every oriented 4-manifold admits spin^c structures. While this fact was proved long ago by Whitney, Hirzebruch, and Hopf, the usual proof is indirect, and rather formal. In this talk, I will first explain a new and more direct proof using twistor spaces. After using these ideas to clarify both old and new results in 4-dimensional geometry, I will go on to explain how this approach can be used to understand both spin and spin^c structures in any dimension.

Roger Penrose (Oxford): Twistors, Pseudo-Twistors, and Split-Octonions

Dave Skinner (Cambridge): Twistors, Integrability and 4d Chern-Simons Theory