Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity

TL;DR

This paper links twistor theory with chiral formulations of General Relativity, especially the pure connection approach, providing new insights and a potential pathway to a twistor action for gravity.

Contribution

It introduces a method to construct complex twistor data from an $SU(2)$-connection, connecting almost Hermitian structures to anti-self-dual-Einstein metrics and proposing new strategies for a twistor action for gravity.

Findings

New proof of the non-linear graviton theorem

Construction of twistor data from $SU(2)$-connections

Discussion of strategies for a twistor action for gravity

Abstract

This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-connection only we show how to construct natural complex data on twistor space, mainly an almost Hermitian structure and a connection on some complex line bundle. Only when this almost Hermitian structure is integrable is the connection related to an anti-self-dual-Einstein metric and makes contact with the usual results. This leads to a new proof of the non-linear-graviton theorem. Finally we discuss what new strategies this "connection approach" to twistors suggests for constructing a twistor action for gravity. In appendix we also review all known chiral Lagrangians for GR.