The foundations of twistor-string theory

The twistor programme of Roger Penrose was conceived as an approach to quantum gravity. In Einstein's theory of general relativity, the geometry of space-time is dynamical and gives rise to the gravitational field. In order to quantize it, one has to consider what kind of background can be taken for granted if geometry itself is to be subject to quantum fluctuations. The twistor programme is to take twistor space to be the given background and the arena in which the theory that unites quantum theory and gravity is most naturally expressed. If this is to work, it must be possible to reformulate all basic physics in terms of structures on twistor space. Early successes were the encoding not only of linear massless fields, but also non-linear gauge fields and gravitational fields with right handed circular polarization. However, until late 2003, this programme had been stuck not only on the problem of encoding the full non-linear structure of Yang-Mills and gravity when they are not circularly polarized, but also of the systematic incorporation of quantum field theory. Witten's introduction of twistor-string theory was a major step forward that now gives a clear idea as to how these longstanding difficulties might be overcome, at least in the context of perturbation theory.The focus of Witten's paper was not on the twistor programme nor quantum gravity, but on finding new mathematical techniques to study gauge theories, the class of theories underlying the force laws of the standard model of particle physics. The perturbative calculation of gauge theory scattering amplitudes is particularly challenging and current analytical and numerical techniques run out of steam at a point below that required by upcoming experiments at the Large Hadron Collider at CERN. Witten's starting point was the remarkable formulae due to Parke and Taylor for so called `maximal helicity violating' (MHV) gauge theory amplitudes. Despite being a sum of many many Feynman diagrams, these formulae are particularly compact. In the late 1980s, Nair had found a remarkable interpretaion of these amplitudes in a supersymmetric context as integrals over a space of holomorphic curves in super twistor space, a supersymmetric extension of Penrose's original twistor space. Witten's generalisation was to express general gauge theory amplitudes as an integral over the space of all algebraic curves, but now of arbitrary genus and degree, in super-twistor space. These formulae have been largely verified (at least at tree level) and have had a substantial impact on perturbative gauge theory. However, this impact is currently limited by a lack of proper understanding of the foundations of twistor-string theory and by the fact that existing twistor-string theories automatically incorporate conformal supergravity, an unphysical theory that necessarily corrupts quantum calculations.In subsequent work of the PI and collaborators, the existence of conformal supergravity was seen as an opportunity because it contains Einstein gravity. Twistor-string theories were constructed in which the gravitational degrees of freedom are precisely those of N=4 and 8 Einstein supergravity. However, these require further investigation as it is not clear whether these predict the correct scattering amplitudes.The aims of this proposal are to explore the underlying geometry, to provide mathematical foundations for the subject and to find and investigate new twistor string theories that can describe Einstein gravity, or even just gauge theories on their own. It has emerged that the correct mathematical framework for understanding these theories involves exciting new ideas from algebraic and differential geometry such as sheaves of chiral algebras and generalised complex structures. The eventual aims are to develop twistor-string theory as a tool for studying quantum gauge theories, and for studying quantum gravity.