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

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.

This is the version published in J.Math.Phys. As compare to the previous version, some paragraph were rewritten to make the text easier to read and some typos corrected

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Cited by in corpus (6)

• Toward a twistor action for chiral higher-spin gravity
• Higher-Spin Self-Dual Yang-Mills and Gravity from the twistor space
• Lectures on twistor theory
• From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras
• Spin current in BF theory
• Yang Mills Theory of Gravity