Subfactors and their connections with low dimensional topology, and low dimensional physics

标题中的“子因素”出现在量子物理学中，作为描述宇宙所需的数学机制的一部分，要求信息的传播速度不能快于光速。子因素有与之相关的数字，可以用来对可能的物理理论进行参数化。其中一个参数被称为“统计维度”。它衡量了当一个系统中的粒子在彼此之间发生变化时所发生的事情的复杂性。如果时空是低维的，这种粒子交换是非常微妙的，因为粒子之间没有相互环绕的空间。这使得与子因素相关的数字在描述时空本身时也是有用的。这些想法被认为为量子计算提供了可能的体系结构。该项目将集中在以下三个主要领域，都与子因素有关：(1)推进平面代数的研究。毕格罗的水母技术是杨-巴克斯特方程的近亲。这项技术的概括应该允许人们将低指标子因子的分类推到指数6，甚至可能更高。(2)低维量子场论。Wassermann指出，WZW共形场理论中的融合运算可以解释为局部可观测代数上双模的Connes融合。最重要的是，这项工作将通过建立西格尔共形场理论的一个明显的么正版本的存在性来扩展。(3)任意平面代数的连续统极限。首席研究员将研究这样一个问题：是否所有的子因子都来自保形场理论？对于这个问题，他首次提出了一种天真的方法--使用平面代数图。让圆上的边界点变得密集是统计力学中的连续极限过程。

The "subfactors" in the title arise in quantum physics as part of the mathematical machinery needed to describe the universe compatibly with the requirement that information can travel no faster than the speed of light. Subfactors have numbers associated with them that can be used to parametrize possible physical theories. One such parameter is known as the "statistical dimension." It measures the complexity of what happens when particles in a system are changed amongst one another. If space-time is low dimensional this particle exchange is very subtle because particles have no room to go around each other. This causes the numbers associated to subfactors to be useful, as well, in the description of space-time itself. Such ideas are thought to provide possible architectures for quantum computing.The project will focus on the following three main areas, all related to subfactors:(1) Advancing research in planar algebras. Bigelow's jellyfish technique is a close cousin of the Yang-Baxter equation. Generaliztions of this technique should allow one to push the classification of low-index subfactors to index 6 and possibly beyond.(2) Low-dimensional quantum field theory. Wassermann showed that the fusion operation in WZW conformal field theory can be interpreted as the Connes fusion of bimodules over algebras of local observables. This work will be extended most importantly by establishing the existence of a manifestly unitary version of Segal conformal field theory. (3) Continuum limits for arbitrary planar algebras. The principal investigator will study the question: Do all subfactors come from conformal field theories? He proposes a first naive approach to this question using the planar algebra picture. Letting the boundary points on the circle become dense is the continuum limit process in statistical mechanics.