New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's Three-Forms

TL;DR

This thesis explores the deep connections between Einstein's gravity in four dimensions, twistor theory, and Hitchin's three-form theories, revealing new geometric insights and relationships across different dimensions.

Contribution

It demonstrates the equivalence of chiral and twistor formulations of gravity and relates four-dimensional gravity to higher-dimensional diffeomorphism-invariant theories of three-forms.

Findings

A new proof of Penrose's non-linear graviton theorem using SU(2)-connections.

Partial encoding of 4D gravity in 7D G2 holonomy manifolds.

Discovery of new topological diffeomorphism-invariant functionals in six dimensions.

Abstract

In this thesis we take Einstein theory in dimension four seriously, and explore the special aspects of gravity in this number of dimension. Among the many surprising features in dimension four, one of them is the possibility of `Chiral formulations of gravity' - they are surprising as they typically do not rely on a metric. Another is the existence of the Twistor correspondence. The Chiral and Twistor formulations might seems different in nature. In the first part of this thesis we demonstrate that they are in fact closely related. In particular we give a new proof for Penrose's `non-linear graviton theorem' that relies on the geometry of SU(2)-connections only (rather than on metric). In the second part of this thesis we describe partial results towards encoding the full GR in the total space of some fibre bundle over space-time. We indeed show that gravity theory in three and four dimensions can be related to theories of a completely different nature in six and seven dimension respectively. This theories, first advertised by Hitchin, are diffeomorphism invariant theories of differential three-forms. Starting with seven dimensions, we are only partially succesfull: the resulting theory is some deformed version of gravity. We however found that solutions to a particular gravity theory in four dimension have a seven dimensional interpretation as G2 holonomy manifold. On the other hand by going from six to three dimension we do recover three dimensional gravity. As a bonus, we describe new diffeomorphism invariant functionnals for differential forms in six dimension and prove that two of them are topological.