Ambitwistor strings and the complex geometry of the S-matrix

Ambitwistor spaces are spaces of complex light-like geodesics in complexified space-times. They were originally introduced by Witten and by Isenberg, Yasskin & Green in 1978 to extend Ward's twistor constructions for self-dual gauge fields to general gauge fields. The ideas extend directly also to gravity. They encode the information of a gauge or gravitational field into the deformation of the complex structure of the ambitwistor space. They therefore convert the analysis of these notoriously difficult partial differential equations into complex analysis. The formulation of the field equations provided in the last century was sufficiently difficult to use that, until last year, the framework had few major applications. However, work of the PI and collaborators has shown that holomorphic string theories in ambitwistor space yield dramatically simpler perturbative formulae for the scattering of gauge and gravitational fields than had hitherto been thought to exist and in particular give the fundamental theory underlying the remarkable formulae of Cachazo, He and Yuan, generating other new remarkably simple formulae also. The simplicity of these formulae is a smoking gun for the existence of a fully nonlinear ambitwistor formulation of gauge and gravity theories that will also reflect this remarkable simplicity despite the lack of complete integrability of these theories. The use of strings allows the reduction of complicated multidimensional complex analysis to the much more tractable complex analysis of Riemann surfaces. The ultimate purpose of this research proposal is to obtain such a fully nonlinear construction that can provide a major new tool for understanding the mathematics and physics of these equations, both classically and quantum mechanically. However, there are many opportunistic projects en route that will generate remarkable new formulae for amplitudes in new contexts, both classical and quantum, and generate new theorems concerning the general structure of the gauge and gravity S-matrices based on complex analysis in ambitwistor space.

Twistor theory has had a substantial impact on algebra, geometry, differential equations and mathematical physics over the last 45 years. The continuing impact on geometry can for example be seen in the recent volume http://www.emis.de/journals/SIGMA/twistors.html.

The developments in this proposal stem from Witten's groundbreaking introduction of twistor-string theory. This gave a new mathematical paradigm for how twistor theory might make contact with physics. It led to a revolution in our understanding of scattering amplitudes and their deep connections with other mathematical structures. The original twistor string was not immediately suitable for physics but the new ambitwistor string overcomes the main obstructions and include Einstein gravity with no requirement for space-time supersymmetry (rather than the maximal conformal gravity of the original model). It gives what must be the correct ways to use ambitwistor space whose applications had hitherto been limited and shows a very deep connection with conventional string theory. There is therefore the likelihood of substantial feedback into standard string theory also.

The simplifications of the ambitwistor formulae for tree amplitudes over their usual Feynman diagram construction are so dramatic that there must be major simplifications in the ambitwistor treatment of fully nonlinear fields sought for in this proposal, in comparison to the usual space-time formulation. This is particularly remarkable in view of the lack of complete integrability of gravity and Yang-Mills and will impact on geometry and analysis. These ideas will be important for mathematical analysis of these equations in the long term.

We will organize a specialized workshop at the end of the second year of the grant to disseminate information and to orient the project for its final year.

Sir Roger Penrose has been an excellent populariser and expositor of science. He was the founder of twistor theory, a subject that he initiated and he also made important early contributions to ambitwistors. He has been very excited by the recent advances and we will invite him to give public lectures on the field.

We will maintain our twistor-theory website so as to make it a valuable resource for the whole field.