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

Understand

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.