**Project**: HigherDimTwistors - Higher-dimensional twistor theory in the language of parabolic geometries

**Person in Charge**: prof. RNDr. Jan Slovák, DrSc

**Host institution**: Faculty of Science, Masaryk University

**Country of Origin**: France

**Project duration**: 24 months

**Scientific panel**: Mathematics

**Abstract**:

The main goal of the project is to expand the already existing strong interdisciplinary research on the borderline of Differential Geometry, Representation Theory, and Mathematical Physics at the Masaryk University. The expertise of Dr. Arman Taghavi-Chabert, coming from the famous Penrose group of Oxford, matches perfectly recent achievements and current projects of the research group around Professor Jan Slovák, with direct links to Rikard von Unge's and Tomáš Tyc's groups in Physics.

More explicitly, the newly established theory of parabolic geometries (Jan Slovák and collaborators) seems to be a great machinery for applications in twistor theory (the so called BGG machinery) and Dr. A. Taghavi-Chabert will bring a unique opportunity for common proceedings in this direction. In particular, the proposal will build on three distinct aspects of twistor theory.

The first of these will be a complete classification of almost optical structures in even dimensions, Lorentzian analogues of almost hermitian structures, and a comprehensive study of their geometric and curvature properties. Examples will be provided, especially in the context of Einstein's field equations.

On a different line, one will develop a twistor model for non-abelian Yang-Mills gerbe theory in six dimensions, which will be of great interest to string theorists. Ultimately, a gerbe-like Penrose-Ward transform will be constructed in this setting.

Finally, a third strand will focus on conformal Killing spinors of higher valence and their geometric prolongation.

**The ongoing project summary: **

The SoMoPro project mostly consists of two distinct parts, yet both make use of twistor theory and parabolic geometry in one way or another.

The first of these is concerned with almost Robinson (or optical) structures on Lorentzian even-dimensional manifolds. From the viewpoint of physics, these may be described as congruences of light rays endowed with complex geometric properties. In four dimensions, these congruences are shearfree. They are Lorenzian analogues of Hermitian structures.

**The initial objectives were to give:**

- a classification of almost Robinson structures in terms of intrinsic torsion classes, similar to the Grey-Hervella classification of almost Hermitian structures;
- an invariant classification of the curvature tensors with respect to an almost Robinson structure, similar to the work of Tricerri, Vanhecke, Falcitelli, Farinola, and Salamon in almost Hermitian geometry;
- and an articulation of the Goldberg-Sachs theorem in higher dimensions.

Given the current interests of general relativists, most of the research has focused on the curvature properties of almost Robinson structures. In particular, Dr. Arman Taghavi-Chabert proved a higher-dimensional generalisation of the Goldberg-Sachs theorem, a fundamental theorem of GR, which relates the degeneracy of the conformal curvature to the integrability of an almost Robinson structure.

In fact, Dr Taghavi-Chabert proved a more general version of the Goldberg-Sachs theorem, which applies to real or complex Riemannian manifolds of any signature and endowed with an almost null structure, i.e. a (complex) distribution by maximal totally degenerate planes.This was done in both even and odd dimensions, and the scope of the project has now in fact been extended to odd-dimensional manifolds as well as even-dimensional ones.

Almost null structures may be viewed as the building blocks of such geometries: a complex conjugate pair of almost null structures defines an almost Hermitian or an almost Robinson structures. For this reason, we have broadened the scope of the project to include the study of almost null structures and their curvature properties. This has led to a spinorial classification of curvature tensors to arbitrary dimensions, thus generalising the notion of principal spinors due to Penrose from four to higher dimensions. From it, it is relatively straightforward to recover a classification of curvature tensors for both almost Hermitian and Robinson manifolds.

There now remains to carry out work towards the classification of the intrinsic torsion classes of Robinson structures. Some results have been obtained so far.

The second part of the project deals with the Penrose-Ward transform in six dimensions. In six dimensions, one has a correspondence between two complex quadrics, which are to be identified with complexified compactified Minkowski space M and its twistor space PT, the space of all self-dual 3-dimensional totally null linear subspaces of M. The Penrose transform establishes an isomorphism between cohomology of sheaves of holomorphic vector bundles over some region Z of PT and solutions to conformally invariant differential equations on a corresponding region X of M.

There are several cases of interest. One is when cohomology on PT gives rise to a 2-form on X, whose exterior derivative is a self-dual 3-form. This 2-form can thus be identified with a gerbe, i.e. a 2-categorical analogue of a connection 1-form on a vector bundle X. The construction is well-known when the structure group of the vector bundle is abelian. However, the ultimate aim of the project would be to develop a non-abelian version of the construction, i.e. a six-dimensional Penrose-Ward transform. The issue of the existence of a non-abelian gauge theory is a difficult one, but Witten conjectured that such a theory exists in a supersymmetric conformal field theory known as the (0,2)-model.

There are however more tractable tangential issues related to six-dimensional twistor theory. The research of Dr Taghavi-Chabert and his collaborators, Prof Lionel Mason and Dr Ron Reid-Edwards, in Oxford, has focussed on the construction of explicit solutions to the massless field equations by means of the Penrose transform and spinor-helicity methods in terms of distributional cohomology classes in the context to massless fields of positive helicity.

Another aspect of this collaborative research has been the investigation of the Xi-transform, a totally real analogue of the Penrose transform introduced by Sparling. This is a six-dimensional version of the four-dimensional X-ray transform, familiar to people working in tomography.

Another direction undertaken by Dr Taghavi-Chabert has been the construction of a Penrose transform for ambichiral theories, i.e. field theories involving massless fields of both positive and negative helicities.This work should pave the way for further development of the Penrose transform in the context of non-conformally invariant field theories, such as six-dimensional Yang-Mills theory.

Both strands of the project are concerned with higher-dimensional generalisations of four-dimensional results, which have proved invaluable in the development of mathematical physics. In the current context of theoretical physics, where extra dimensions of spacetime are thought to play a fundamental rôle, this project instils novelty and originality to the field.

The classification of the conformal tensor in four dimensions has historically played a huge rôle in the discovery of solutions to Einstein’s field equations, to which the Goldberg-Sachs theorem has been instrumental. Hence, generalizing these ideas to higher dimensions will provide deeper insight into the geometry of the universe.

Similarly,six-dimensional twistor theory provides a powerful and elegant formalism, by means conformal field theories can be expressed. Time and again, twistor theory in four dimensions has simplified the computation of scattering of fundamental particles, and it is highly expected to achieve the same result in six or even higher dimensions.

Thus, the work carried out as part of this SoMoPro project will bring the South Moravian region to the forefront of fundamental research.