The Haagerup subfactor, K-theory and conformal field theory

算子代数理论是由默里和冯·诺伊曼提出的，作为研究群表示的工具和量子力学的数学框架。从那时起，非交换算子代数已成为分析奇异拓扑空间和测量空间（如叶形的叶空间或遍历群作用的轨道空间）的有力工具，并应用于数学和理论物理的许多领域，例如K理论，诺维科夫猜想，叶形，连杆和3流形不变量，量子群，模不变量，拓扑和共形场论，以及d膜。随着Connes引入非交换几何，以及时空的非交换模型，开辟了全新的研究途径。非交换环面，通过其K理论，已经被用来提供量子场论中的矩阵模型。Jones的子因子理论导致了与连杆和三流形不变量、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。该方案实际上将这两个领域联系起来-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们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学的语言。非平凡量子场论是最简单和最对称的，因此是最适合数学研究的，应该是所谓的二维时空共形场论CFT。这些cft本身对物理学有直接的兴趣，最著名的是因为他们的研究相当于摄动弦理论。将Wightman公理应用于CFT，可以非常直接地得到顶点算子代数VOA的定义，这是由Borcherds首先表述的。VOA可以被看作是量子场论中最对称的数学上严格的代数表述。所有已知的cft都是应用于标准例子的简单结构的结果。但子因子的分类导致了外来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维拓扑场论将数与封闭的n维流形联系起来，将向量空间与封闭的n-1维流形联系起来。当n=3时，与环面相关联的空间具有环形结构- Verlinde环。通过将一个线性范畴附加到闭合的n-2维流形上，可以扩展Atiyah的定义；当n=3时，圆与捕获eg的模张量类别相关联。编织组和模组表示。FHT对Verlinde环的k理论解释的一个自然框架是降维，这是一种从n维拓扑场理论到n-1维拓扑场理论（更对称）的标准方法；将其应用于n=3和圆，得到Verlinde环。所有这一切的一些类似应该适用于任何CFT，探索这是我们建议的一部分。FHT只考虑CFT的手性一半；两个手性半部分以不同寻常的方式拼接在一起，由此产生的完整CFT在数学上非常丰富——参见Evans及其合作者的编织子因子方法，或Fuchs等人对完整CFT的分类解释。我们的大部分项目都是为了填补这一空白，并了解FHT如何扩展到完整的CFT。从这个意义上说，我们正试图揭示最简单的非平凡量子场论的基本结构。本项目旨在解释和构建CFT，包括超越有限和循环群数据的奇异模型，使用子因子，扭曲k理论和VOA的工具。

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.