It’s becoming clear what the hot new topic in particle theory is these days: the use of twistor space methods to try and understand scattering amplitudes in Yang-Mills and gravity theories, especially the maximally supersymmetric versions. This evening on the arXiv there are two closely related papers on the topic: Scattering Amplitudes and BCFW Recursion in Twistor Space by Lionel Mason and David Skinner, and The S-Matrix in Twistor Space, by Arkani-Hamed and collaborators (there’s also a third, much more distantly related paper, this one). The Arkani-Hamed et al. paper gives an extensive discussion of motivation for this work in the introduction: the structure of scattering amplitudes for these theories is remarkably simple in twistor space, leading to the question of whether one can formulate the full theory directly in twistor space somehow, giving a different sort of holographic dual than the AdS/CFT one.
A very influential version of this idea goes back to Witten’s 2003 paper Perturbative Gauge Theory as a String Theory in Twistor Space, where the dual theory investigated was a topological string theory. This most recent work doesn’t appear to use topological string theory, although Arkani-Hamed et al. are rather cagey on the topic of what sort of twistor space theory is at issue. They promise a forthcoming paper entitled “Holography and the S-matrix”, with:
a completely different picture for computing scattering amplitudes at tree level than given by the BCFW formalism, that we strongly suspect is connected with a maximally holographic description of tree amplitudes that makes all the symmetries of the theory manifest but completely obscures space-time locality.
The history of using twistor-space to study gauge theory goes back a very long ways. Penrose started using twistor techniques to study gravity back in the mid-sixties, and after the 1975 discovery of self-dual solutions to the YM equations (instantons), it became apparent that twistor-space techniques could be used to solve them, turning the problem of solving these non-linear equations into a problem in algebraic geometry, that of constructing certain holomorphic vector bundles. Atiyah was among the mathematicians who got interested in this, and his beautiful 1979 lecture notes Geometry of Yang-Mills Fields remain a wonderful introduction to the mathematical side of the subject. While still a post-doc, Witten worked in this area, coming up in 1978 with an interpretation of the full YM equations using supersymmetry. Work on this topic was what brought Atiyah and other mathematicians into contact with physicists, including Witten, beginning a quite remarkable period of very successful interaction between the two camps.
Twistors were among the things I started thinking about at the end of my graduate school days in the mid-eighties, for a completely different reason, one that has nothing to do with the current interest in the subject. I was interested in the problem of how to put spinors on a lattice, and the twistor geometry story gives a beautiful way of thinking geometrically about spinors. This idea never really got anywhere, although I did notice some relations between the geometry of the standard model gauge groups and representations that I wrote about back in 1987 and still find quite remarkable. It will be interesting to see what new ideas emerge from this latest wave of interest in thinking about quantum field theory in twistor-space.