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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

基本信息

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

项目摘要

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年代早期，Vaughan Jones将他的子因子理论引入到冯·诺依曼算子代数的分析中。这个理论与拓扑学和几何学中的纽结理论、统计力学和共形量子场论有着深刻的联系。子因子编码统计力学模型的对称性，或者在临界温度下编码共形量子场论的对称性。维数在数学中无处不在。琼斯指数测量较大代数相对于较小代数的相对维数。K-理论也提供了工具来理解尺寸和这两个概念是在这个项目的心脏。人们认为，异国情调的模型可以产生的子因子框架超出了标准模型的基础经典群对称性。然而，制作这样的模型被证明是难以捉摸的。在1993年乌夫Haagerup产生了一个候选人的异国情调的子因子的不合理的方面，即琼斯指数是一半的5加上平方根的13 -基本上生产的玻尔兹曼重量在临界状态下，由他赤手空拳与强大的可积性。然而，埃文斯和甘农已经证明，人们可以从基本有限群的经典对称性、三个物体的对称性和13维正交变换群中理解这个子因子，从而为基础的共形量子场论可以用共形网或顶点算子代数来描述提供了证据。他们还提供了证据表明，Haagerup家庭属于一个无限的家庭，最近发现了另一个相关系列的二次系统，再次强有力的证据基础上的基本经典对称性的基础共形场理论。这个计划是分析和阐明这些神秘的子因子和二次系统，Haagerup系统和相关系列，被认为是无限的，近群系统，它们的普遍推广为二次系统。这些工具来自算子代数，特别是子因子，共形网和扭曲等变K理论。目标是实现其对偶或对称包络代数的基本模张量范畴和恢复二次系统。