The Haagerup subfactor, K-theory and conformal field theory

算符代数理论是由默里和冯·诺依曼作为研究群表示的工具和量子力学的数学框架而创立的。从此，非交换算子代数成为分析奇异拓扑空间和测度空间的有力工具（如叶空间的叶状或轨道空间的遍历群作用）与链接和应用的许多领域的数学和理论物理，如K理论，诺维科夫猜想，叶状，链接和3流形不变量，量子群，模不变量，拓扑和共形场论，以及D膜。随着康纳斯对非对易几何的引入，以及时空的非对易模型的提出，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，已经被用来提供量子场论中的矩阵模型。子因子理论的琼斯导致连接和3流形不变量，量子群，完全可解模型的统计力学，共形量子场论和模块化不变量。该计划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。艾德维滕预言，21世纪世纪数学的一个主题将是数学家们接受量子场论，这是物理学家们认为描述基础物理学的语言。最简单、最对称、因而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。这些CFTs本身就是物理学的直接兴趣，最著名的是因为它们的研究等同于微扰弦理论。应用Wightman公理CFT导致非常直接的顶点算子代数VOA的定义首先制定的Borcherds。VOA可以被看作是最对称的量子场论的数学上严格的代数公式。所有已知的CFTs都是应用于标准示例的直接构造的结果。但是子因子的分类导致了外来CFTs的候选者。我们项目的一部分就是探索这些可能性。阿蒂亚告诉我们，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。对于n=3，与环面相关联的空间具有环结构-Verlinde环。Atiyah的定义可以通过将线性范畴附加到封闭的n-2维流形上来扩展;对于n=3，圆与捕获例如的模张量范畴相关联。辫子群和模群表示。FHT对Verlinde环的K-理论解释的一个自然框架是降维，这是从n维拓扑场论到n-1维拓扑场论（具有更多对称性）的标准方法;应用于n=3和圆，它产生了Verlinde环。所有这一切的一些类似物应该适用于任何CFT，探索这一点是我们建议的一部分。FHT只考虑CFT的手性一半;两个手性一半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-参见例如Evans和合作者的编织子因子方法的工作，或者例如Fuchs等人的完整CFT的分类解释。我们的大部分项目是为了填补这一空白，并了解FHT如何扩展到完整的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的，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.