Holomorphic Linking and the Twistor Geometry of the S-matrix

全纯连接是Michael Atiyah爵士在1981年引入的，作为三维空间中封闭曲线的高斯连接数的复杂模拟。他是在试图在四球上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。结理论和三流形拓扑随后被引入非阿贝尔连杆不变量，如琼斯多项式，这是与某些三维规范理论（陈-西蒙斯理论）相关的“威尔逊循环”，彻底改变了。量子场论的s矩阵是包含所有可能散射过程的所有可用数据（振幅）的数学对象。最近，在四维时空的某些规范理论中，s矩阵的扭转几何得到了广泛的研究，并且发现其内容非常丰富，涉及到许多不同的几何思想，从曲线模空间上的积分到格拉斯曼周期的研究。这在去年达到了顶峰，人们认识到s矩阵最好被理解为扭曲空间中复杂多边形的全纯连杆不变量，它对s矩阵所依赖的数据进行编码。尽管全纯连杆不变量在30年前就被提出了，但可用的数学技术仍然处于初级阶段，事实上，这个s矩阵是第一个被定义和研究的非abel例子。本提案旨在将全纯连接技术发展成一个框架，该框架可用于解决全s矩阵，并将其思想扩展到其他相关问题。解决方案将需要基于格拉斯曼积分公式的多对数的新颖几何构造。它还需要研究非线性可积方程组（某些希钦系统）及其量化。

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.