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The Haagerup subfactor, K-theory and conformal field theory

Haagerup 子因子、K 理论和共形场论

基本信息

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

项目摘要

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.

算符代数理论是由Murray和von Neumann首创的，作为研究群表示的工具和量子力学的数学框架。从那时起，非交换算子代数已经成为分析奇异拓扑空间和测度空间(如叶层的叶空间或遍历群作用的轨道空间)的有力工具，并与数学和理论物理的许多领域建立了联系并得到了应用，如K理论、Novikov猜想、叶层、链接和3-流形不变量、量子群、模不变量、拓扑场和共形场论以及D-膜。随着康纳斯引入非对易几何，以及随之而来的非对易时空模型，开辟了一条全新的研究途径。非对易环面通过他们的K理论被用来提供量子场论中的矩阵模型。琼斯的子因子理论导致了与链式和3-流形不变量、量子群、统计力学中的精确可解模型、共形量子场论和模不变量的联系。这个程序实际上将这两个领域联系在一起--利用Freed-Hopkins-Teleman Fht将Verlinde环实现为扭曲等变K理论，以及Wassermann等人将Verlinde环实现为循环群子因子的自同态或双模。Ed Witten预测，21世纪数学的一个主题将是数学家与量子场论达成协议，物理学家认为量子场论是描述基础物理的语言。最简单、最对称、最适合数学研究的非平凡量子场论应该是所谓的二维时空共形场论CFT。这些CFT本身在物理学中有直接的意义，最著名的是因为它们的研究等同于微扰弦理论。将Wightman公理应用于CFT，非常直接地引出了顶点算子代数VOA的定义，该定义首先由BorCherds提出。VOA可以被看作是最对称的量子场论的数学严谨的代数公式。所有已知的CFT都是应用于标准示例的直接构造的结果。但子因素的分类导致了奇异CFT的候选者。我们项目的一部分就是探索这些可能性。Atiyah告诉我们，n维拓扑场论将数与n维闭流形联系起来，将向量空间与n-1维闭流形联系起来。当n=3时，与环面相关的空间有一个环结构--Verlinde环。Atiyah的定义可以被扩展，通过将线性范畴附加到n-2维闭流形上；对于n=3，圆与模张量范畴相关联，例如。辫子群和模群表示。FHTK理论解释Verlinde环的一个自然框架是降维，这是从n维拓扑场理论到n-1维拓扑场理论(具有更多对称性)的标准方法；应用于n=3和圆，它产生Verlinde环。所有这些都应该适用于任何CFT，探索这一点是我们建议的一部分。FHT只考虑CFT的手性部分；两个手性部分以非平凡的方式拼接在一起，得到的完整CFT在数学上是非常丰富的-参见例如Evans和合作者的工作以获得辫子因子方法，或例如Fuchs等人的工作以获得完整CFT的分类解释。我们的项目很大程度上是为了填补这一空白，并了解FHT如何延伸到完整的CFT。在这个意义上，我们试图揭示最简单的非平凡量子场论的基本结构。这个项目开始解释和构建CFT，包括有限和环群数据之外的奇异模型，使用子因子、扭曲K理论和VOA的工具。