Holomorphic Linking and the Twistor Geometry of the S-matrix

全纯链接是由Michael Atiyah爵士于1981年提出的，它是三维空间中闭曲线的Gauss链接个数的复类。他是在试图利用扭曲空间的复杂几何构造拉普拉斯函数时得出这个结论的，扭曲空间是一个复杂的三维空间。纽结理论和三种流形拓扑随后被引入非阿贝尔链接不变量，例如琼斯多项式，它们是与某些三维规范理论(Chern-Simons理论)相关的威尔逊环。量子场论的S矩阵是包含所有可能的散射过程的所有可用数据(振幅)的数学对象。最近，关于四维时空中某些规范理论的S矩阵的扭曲几何得到了深入的研究，并发现其非常丰富，这与许多不同的几何思想相联系，从曲线的模空间上的积分到对草人循环的研究。这在去年达到了顶峰，人们意识到S矩阵应该最好地理解为扭曲空间中复杂多边形的全纯链接不变量，它编码了S矩阵所依赖的数据。尽管全纯链接不变量是在30年前提出的，但可用的数学技术仍然处于初级阶段，事实上，这个S矩阵是定义和研究的第一个非阿贝尔例子。这一建议旨在将全纯链接的技术发展成一个框架，可以用于求解完整的S矩阵，并将其思想扩展到其他相关问题。解将需要基于Grassman积分公式的多对数的新的几何结构。它还需要研究非线性可积方程组(某些希钦系统)及其量子化。

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.