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

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.

20 世纪 80 年代初，Vaughan Jones 在算子冯诺依曼代数分析中引入了他的子因子理论。此后，该理论与拓扑和几何、统计力学和共形量子场论中的纽结理论建立了深厚的联系。子因子编码统计力学模型的对称性或临界温度下的共形量子场论。维度在数学中无处不在。琼斯指数衡量较大代数相对于较小代数的相对维数。 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，除了具有基础经典群对称性的标准模型之外，还可以在子因子框架中生成奇异模型。然而，生产这样的模型被证明是难以捉摸的。 1993 年，Uffe Haagerup 提出了一个非理性维度的奇异子因子的候选因子，即琼斯指数是 5 的一半加上 13 的平方根 - 本质上是通过他徒手产生具有强可积性质的临界玻尔兹曼权重。然而 Evans 和 Gannon 已经表明，人们可以从基本有限群的经典对称性、三个物体的对称性以及 13 维正交变换群来理解这个子因子，从而为底层共形量子场论提供了证据，该理论可以通过共形网络或顶点算子代数。他们还提供了证据，证明哈格鲁普家族属于无限家族，并且最近发现了另一个相关的二次系统系列，再次提供了基于基础经典对称性的基础共形场论的有力证据。该计划旨在分析和阐明这些神秘的子因子和二次系统，哈格鲁普系统和相关系列，被认为是无限的，近群系统，它们的共同概括为二次系统。这些工具来自算子代数，特别是子因子、共形网络和扭曲等变 K 理论。目标是实现其双精度或对称包络代数的底层模张量类别以及二次系统的恢复。

项目成果

Classification of module categories for SO(3)2

SO(3)2 模块类别的分类

• DOI: 10.1016/j.aim.2021.107713
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Evans D

Classification of Module Categories for $SO(3)_{2m}$

$SO(3)_{2m}$ 的模块类别分类

• DOI: 10.48550/arxiv.1804.07714
• 发表时间: 2018
• 作者: Evans D

Reconstruction and local extensions for twisted group doubles, and permutation orbifolds

• DOI: 10.1090/tran/8575
• 发表时间: 2018-04
• 期刊: arXiv: Quantum Algebra
• 作者: David E. Evans;T. Gannon

Spectral measures for G2, II: Finite subgroups

G2, II 的谱测度：有限子群

• DOI: 10.1142/s0129055x20500269
• 发表时间: 2020
• 期刊: Reviews in Mathematical Physics
• 影响因子: 1.8
• 作者: Evans D

Equivariant higher twisted K-theory of SU(n) for exponential functor twists

指数函子扭曲的 SU(n) 等变高扭曲 K 理论

• DOI: 10.48550/arxiv.1906.08179
• 发表时间: 2019
• 作者: Evans D