Holomorphic Linking and the Twistor Geometry of the S-matrix

Holomorphic linking was introduced by Sir Michael Atiyah in 1981 as a complex analogue of the Gauss linking number of closed curves in three dimensional space. He was led to this whilst trying to construct Green's functions for the Laplacian on the four-sphere using the complex geometry of twistor space, a complex three-dimensional space. Knot theory and three manifold topology were subsequently revolutionized by the introduction of nonabelian link invariants such as the Jones polynomial, which are `Wilson loops' associated to certain three-dimensional gauge theories (Chern-Simons theories).

The S-matrix of a quantum field theory is the mathematical object that contains all the available data (amplitudes) of all possible scattering processes. Recently the twistor geometry of the S-matrix for certain gauge theories on four-dimensional space-time has come under intense study and found to be extraordinarily rich, making contact with many different geometrical ideas ranging from integrals over moduli spaces of curves to the study of cycles in grassmannians. This culminated last year with the realization that the S-matrix should best be understood as a holomorphic link invariant of a complex polygon in twistor space that encodes the data on which the S-matrix depends.
Although holomorphic link invariants were proposed 30 years ago, the available mathematical technology remains rudimentary and indeed this S-matrix is the first nonabelian example to be defined and studied. This proposal seeks to develop the technology underlying holomorphic linking into a framework that can be used to solve for the full S-matrix and to extend the ideas to other related problems. The solution will require novel geometrical constructions of polylogarithms based on Grassmannian integral formulae. It will also require the study of nonlinear integrable systems of equations (certain Hitchin systems) and their quantization.

The immediate beneficiaries of this research will be mathematicians working in complex and algebraic geometry and theoretical physicists with an interest in geometric tools for calculating the S-matrix and correlation functions.

Complex and algebraic geometers will have new invariants for studying varieties containing appropriate curves that link directly into other tools arising from holomorphic Chern-Simons theories such as Donaldson-Thomas invariants. Theoretical physicists will have new tools for computing the S-matrix for gauge theories and gravity that are dramatically simpler than standard Feynman diagrams. This will impact on theoreticians who compute cross sections for collider physicists in CERN and other particle accelerators.

Communication and engagement
Communication will be by advertised seminars, both in university departments and in conferences nationally and internationally and personal interaction. We request funds for the PDRA to attend one national and one international conference per year, and for the PI two attend two over the course of the project. Papers will be posted on the arxiv and submitted to the leading journals in the field. We will also organize an informal meeting at the end of the second year of the project to take stock of progress and to prioritize the research in the final year.
We also plan one visit per year to the IAS at Princeton and to the Perimeter Institute in Canada to cross-fertilize with the programmes of other leading groups in the field.

Collaboration
The collaboration Dr Skinner will be continued by video-conferencing and by a yearly trip to the Perimeter Institute.