The search for the exotic : subfactors, conformal field theories and modular tensor categories

寻找奇异的东西：子因子、共形场论和模张量类别

• 批准号: EP/N022432/1
• 负责人: David Evans
• 金额: $ 44.04万
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN022432%2F1
• 关键词: search; exotic; subfactors; conformal; field; search; exotic; subfactors; conformal; field

In the early 1980's, Vaughan Jones introduced his subfactor theory in the analysis of von Neumann algebras of operators. This theory has since found deep connections with knot theory in topology and geometry, statistical mechanics and conformal quantum field theory. A subfactor encodes the symmetry of a statistical mechanical model or at the critical temperature a conformal quantum field theory. Dimension is ubiquitous in mathematics. The Jones index measures the relative dimension of the larger algebra over the smaller. K-theory also provides tools for understanding dimension and both of these notions are at the heart of this project.It was thought that exotic models could be produced in the subfactor framework beyond the standard models which had underlying classical group symmetries. However producing such models proved elusive. In 1993 Uffe Haagerup produced a candidate for an exotic subfactor of irrational dimension, namely the Jones index is half of 5 plus the square root of 13 - by essentially producing the Boltzmann weights at criticality by his bare hands with strong integrability properties. However Evans and Gannon have shown that one can understand this subfactor from classical symmetries of an elementary finite group, the symmetries of three objects, and the group of orthogonal transformations in 13 dimensions, producing evidence for an underlying conformal quantum field theory to be described by a conformal net or vertex operator algebra. They have also provided evidence that the Haagerup family belongs to an infinite family and most recently found another related series of quadratic systems, with again strong evidence of underlying conformal field theories based on underlying classical symmetries.This programme is to analyse and shed light on these mysterious subfactors and quadratic systems, the Haagerup systems and related series, thought to be infinite, the near group systems, their common generalizations as quadratic systems. The tools are from operator algebras, particularly subfactors, conformal nets and twisted equivariant K-theory. The objectives are the realization of the underlying modular tensor categories of their doubles or symmetric enveloping algebras and the recovery of the quadratic systems.

在1980年代初期，沃恩·琼斯（Vaughan Jones）在分析运营商的冯·诺伊曼（Von Neumann）代数时介绍了他的子因素理论。此后，该理论在拓扑和几何学，统计力学和保形量子场理论中与结理论的联系很深。亚比例编码统计机械模型的对称性或在临界温度A量子量子场理论。维度在数学中无处不在。琼斯指数测量较小代数的相对维度。 K理论还提供了理解维度的工具，这两个概念都是该项目的核心。人们认为，除了具有基本的经典群体对称性的标准模型之外，可以在亚比例框架中产生异国模型。但是，生产这种模型被证明是难以捉摸的。 1993年，Uffe Haagerup生产了一个异国情调的非理性维度亚基特的候选者，即琼斯指数是5的一半加13的平方根 - 基本上是通过他裸露的手与强大的可集成属性产生批判性的鲍尔茨曼重量。然而，埃文斯（Evans）和甘农（Gannon）表明，一个人可以从基本有限群的经典对称性，三个对象的对称性以及13个维度的正交转换组中理解这一亚比例，从而产生了基本的量子量子理论的证据。他们还提供了证据表明，哈格普家族属于一个无限的家庭，最近发现了另一个相关的二次系统，并再次有强有力的证据表明，基于基本的经典对称性基于基础的保质野外理论，该计划将分析并揭示这些神秘的子接地仪和Quadratic Systems，众所周知的系统，与他们相关的系统，与他们相关的系统，与他们相关的系统，近距离构成了这些系统，这些系统是近距离的，这些系统是近距离的。系统。这些工具来自操作员代数，尤其是子因子，保形网和扭曲的Epovariant K理论。这些目标是实现其双打或对称包络代数的基础模块化张量类别以及二次系统的恢复。