Subfactors and conformal field theory

在这个项目中，首席研究员将研究子因子和共形场理论之间的相互作用，这是一个最近被证明非常富有成果的主题。本研究的重点放在本课题的两个方面：(1)使用算子代数技术研究超对称和奇异共形场理论；(2)研究从有限群论到子因子一般集合的猜想的广泛推广。这些算子代数技术的应用有望为许多表示理论问题提供答案，并阐明某些奇异子因子，例如由保角包含引起的子因子。子因子的一般框架也将为一些旧的群论猜想提供新的见解。算子代数理论是由约翰·冯·诺伊曼提出的，目的是为量子力学提供一个合适的数学框架。非交换性（即乘积AB可能不同于BA）是量子力学的一个关键特征，也是算子代数的一个重要方面。Vaughan Jones的子因子理论就是建立在这个非交换框架上的，它为算子代数的联系提供了数学的一些主要部分。共形场论是凝聚态物理中描述临界现象的理论，在弦理论中也占有重要地位。子因子和共形场论之间的显著相互作用导致了许多有趣的数学发展。这项研究的目的是为在这种背景下出现的一些重要的数学问题找到解决方案，这些问题具有广泛的应用。

In this project the principal investigator will study interactions between subfactors and conformal field theories, a topic that has proved to be very fruitful lately. The main emphasis of this research is placed on two aspects of the subject: (1) the study of supersymmetric and exotic conformal field theories using operator algebraic techniques, and (2) the study of a vast generalization of a conjecture from finite group theory to the general setting of subfactors. The applications of these operator algebraic techniques are expected to produce answers to a number of representation theory questions and to shed light on certain exotic subfactors, such as those arising from conformal inclusions. The general framework of subfactors will also provide new insight into some old group theory conjectures. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework for quantum mechanics. Noncommutativity (i.e., the fact that a product AB may differ from BA), which is a key feature of quantum mechanics, is an important aspect of operator algebras. Vaughan Jones's subfactor theory is built on this noncommutative framework and has provided linkages of operator algebras with some major parts of mathematics. Conformal field theory is a theory describing critical phenomena in condensed matter physics, and it also plays an important role in string theory. The remarkable interactions between subfactors and conformal field theory have led to many interesting mathematical developments. The aim of this research is to find solutions to some of the important mathematical issues that surface in this context and that have a wide range of applications.