The Haagerup subfactor, K-theory and conformal field theory

Haagerup 子因子、K 理论和共形场论

• 批准号: EP/J003352/1
• 负责人: David Evans
• 金额: $ 6.1万
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ003352%2F1
• 关键词: Haagerup; subfactor; theory; conformal; field

The theory of operator algebras was initiated by Murray and von Neumann as a tool for studying group representations and as a mathematical framework for quantum mechanics. Since then noncommutative operator algebras have become a powerful tool in the analysis of singular topological spaces and measure spaces (such as leaf spaces of foliations or orbit spaces of ergodic group actions) with links and applications to many areas of mathematical and theoretical physics, eg K theory, the Novikov conjecture, foliations, link and 3-manifold invariants, quantum groups, modular invariants, topological and conformal field theory, and D-branes. With the introduction by Connes of noncommutative geometry, and with it noncommutative models of space-time, entirely new avenues of research have opened up. Noncommutative tori, through their K theory, have been used to provide matrix models in quantum field theory. The subfactor theory of Jones led to connections with link and 3-manifold invariants, quantum groups, exactly solvable models in statistical mechanics, conformal quantum field theory and modular invariants. This programme actually links these two areas - utilising the realisation of the Verlinde ring by Freed-Hopkins-Teleman FHT as twisted equivariant K-theory, and the realisation by Wassermann et al of the Verlinde ring as endomorphisms or bimodules of loop group subfactors. Ed Witten predicted that a theme of 21st century mathematics will be mathematicians coming to terms with quantum field theory, which is the language physicists believe describes fundamental physics. The nontrivial quantum field theories which are the simplest and most symmetrical, and so are the most amenable to mathematical study, should be the so-called conformal field theories CFT in 2 space-time dimensions. These CFTs are themselves of direct interest in physics, most famously because their study is equivalent to that of perturbative string theory. Applying the Wightman axioms to CFT leads very directly to the definition of a vertex operator algebra VOA first formulated by Borcherds. A VOA can be regarded as a mathematically rigorous algebraic formulation of the most symmetric of the quantum field theories. All known CFTs are the results of straightforward constructions applied to standard examples. But the classification of subfactors leads to candidates for exotic CFTs. Part of our project is to explore these possibilities. Atiyah tells us that an n-dimensional topological field theory associates numbers to closed n-manifolds and vector spaces to closed n-1 dimensional manifolds. For n=3, the space associated to a torus comes with a ring structure - the Verlinde ring. Atiyah's definition can be extended, by attaching a linear category to closed n-2 dimensional manifolds; for n=3, the circle is associated to a modular tensor category capturing eg. the braid group and modular group representations. A natural framework for FHT's K-theoretic interpretation of the Verlinde ring is dimensional reduction, a standard way to go from an n-dimensional topological field theory to say an n-1 dimensional one (with more symmetry); applied to n=3 and the circle, it yields the Verlinde ring. Some analogue of all this should apply to any CFT, and exploring this is part of our proposal. FHT only consider a chiral half of the CFT; the two chiral halves splice together in nontrivial ways, and the resulting full CFT is very rich mathematically - see eg the work of Evans and collaborators for a braided subfactor approach, or eg Fuchs et al for categorical interpretation, of the full CFT. Much of our project is to fill this gap, and understand how FHT extends to the full CFT. In this sense we are trying to uncover the basic structure of the simplest nontrivial quantum field theories.This project sets out to explain and construct CFT's, including exotic models beyond finite and loop group data, using tools from subfactors, twisted K-theory an VOA's.

默里（Murray）和冯·诺伊曼（Von Neumann）启动了操作者代数理论，作为研究小组表示的工具，也是量子力学的数学框架。 Since then noncommutative operator algebras have become a powerful tool in the analysis of singular topological spaces and measure spaces (such as leaf spaces of foliations or orbit spaces of ergodic group actions) with links and applications to many areas of mathematical and theoretical physics, eg K theory, the Novikov conjecture, foliations, link and 3-manifold invariants, quantum groups, modular invariants, topological and共形场理论和D-BRANES。随着非交通性几何形状的引入，并随着时空的非共同模型的引入，全新的研究途径就开始了。通过其k理论，非交换性托里已被用来在量子场理论中提供基质模型。琼斯的子因子理论导致与链接和3个manifold不变性，量子组，统计力学中的确切可解决模型，保形量子场理论和模块化不变性的联系。该程序实际上将这两个领域链接起来 - 利用Freed-Hopkins-Teleman FHT实现Verlinde环为扭曲的Equivariant K理论，以及Verlinde环的Wassermann等人的实现作为内态性或循环组亚曲线的内态性或biModules。埃德·维滕（Ed Witten）预测，21世纪数学的主题将是数学家与量子场理论相处，这是物理学家认为描述的基本物理学。非平凡的量子场理论是最简单，最对称的，也是最适合数学研究的非平凡量子场理论，应该是2个时空维度中所谓的保形场理论CFT。这些CFT本身就是物理学的直接兴趣，最著名的是它们的研究等同于扰动弦理论。将Wightman Axioms应用于CFT，非常直接导致了Borcherds首先制定的顶点操作员代数VOA的定义。 VOA可以被视为量子场理论中最对称的数学严格代数公式。所有已知的CFT是应用于标准示例的直接结构的结果。但是，子因子的分类导致候选外来CFT。我们项目的一部分是探索这些可能性。 Atiyah告诉我们，N维拓扑场理论将数字与封闭的N-manifolds和向量空间相关联，与封闭的N-1维流形相关。对于n = 3，与圆环相关的空间带有环结构 - verlinde环。可以通过将线性类别附加到封闭的N-2维歧管上，可以扩展Atiyah的定义。对于n = 3，圆与捕获的模块化张量类别相关联。编织组和模块化组表示。 FHT对Verlinde环的K-k理论解释的自然框架是尺寸降低，这是一种从n维拓扑场理论进行的标准方法，可以说是N-1维数（具有更多的对称性）；应用于n = 3和圆，它产生了Verlinde环。所有这些的类似物都适用于任何CFT，探索这是我们建议的一部分。 FHT仅考虑手性的一半。这两个手性一半的剪接以非平凡的方式拼接在一起，并且由此产生的完整CFT在数学上非常丰富 - 例如，请参见Evans和合作者的工作，以进行编织的亚比例方法，或者例如Fuchs等用于分类解释，即完整CFT。我们的大部分项目都是填补这一空白，并了解FHT如何扩展到完整的CFT。从这个意义上讲，我们试图揭示最简单的非平凡量子场理论的基本结构。此项目着手使用来自有限和循环组数据的外来模型来解释和构建CFT，其中包括来自有限和循环组数据的外来模型，使用子因子的工具，扭曲的K理论AN VOA。

项目成果

Near-group fusion categories and their doubles

• DOI: 10.1016/j.aim.2013.12.014
• 发表时间: 2012-08
• 期刊: arXiv: Quantum Algebra
• 作者: David E. Evans;T. Gannon

Non-unitary fusion categories and their doubles via endomorphisms

非酉融合类别及其通过自同态的双打

• DOI: 10.1016/j.aim.2017.01.015
• 发表时间: 2017
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Evans D

Modular Invariants and Twisted Equivariant K-theory III: KK-theory

模不变量和扭曲等变 K 理论 III：KK 理论

• 期刊: to be submitted in April 2016 after careful proofreading
• 作者: Evans DE