Subfactors from Coset Conformal Field Theories

摘要：徐锋的研究涉及子因子理论，这是算子代数领域的一个分支，对数学的各个领域，特别是对低维拓扑产生了巨大的影响。利用仿射Kac-Moody代数的表示理论和代数量子场论的思想，可以构造一类称为Jones-Wassermann子因子的子因子。这些子因子与备受关注的二维共形场理论有着密切的关系。本项目的目的是研究与共形场理论相关的琼斯-瓦瑟曼子因子，这在理论上耗尽了一大批共形场理论。提出的研究问题的解决方案，虽然集中在子因子上，但预计将导致与无限维代数的表示理论和低维拓扑相结合的开放性问题的答案。本世纪初出现的量子力学彻底改变了我们对物理世界的理解，它提出了许多重要的数学（更不用说哲学）问题。算子代数理论是由约翰·冯·诺伊曼提出的，目的是提供一个适当的数学框架来处理这类问题。子因子理论是算子代数学科的一个分支，研究较小的代数在较大的代数中可能占据的“位置”。事实证明，由于所讨论的代数的复杂性质，可能的位置受到相当严格的控制，允许的配置通常反映了代数所对应的量子力学系统的对称性。对称性，特别是子因子理论帮助揭示的隐藏的对称性，在科学中扮演着重要的角色，因为它们通常提供了通过将必须考虑的自由参数的数量纳入数学上可管理的范围来降低所研究问题复杂性的关键。因此，寻找隐藏的对称性已成为许多物理科学家活动的焦点。子因子理论为研究人员提供了一种精确的工具来理解数学和物理环境中的对称性。这个项目的目的是阐明在这种背景下出现的一些重要的数学问题。

Proposal: DMS-9820935Principal Investigator: Feng XuAbstract: Xu's research deals with subfactor theory, a branch of the field of operator algebras that has had an enormous impact on various areas of mathematics, above all on low-dimensional topology. By using the representation theory of affine Kac-Moody algebras and ideas from algebraic quantum field theory, a class of subfactors known as Jones-Wassermann subfactors can be constructed. These subfactors have close relations to two-dimensional conformal field theories that have attracted great attention. The aim of this project is to study Jones-Wassermann subfactors related to coset theories, which conjecturally exhaust a large class of conformal field theories. Solutions to the problems proposed for investigation, though focused on subfactors, are expected to lead to answers to open questions arising in conjunction with both the representation theory of infinite dimensional algebras and low-dimensional topology. Quantum mechanics, which revolutionized our understanding of the physical world when it arrived on the scene early in this century, raised a number of significant mathematical (not to mention philosophical) questions. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework in which to deal with such questions. Subfactor theory is a branch of the subject of operator algebras that studies the "positions" a smaller algebra might occupy within a larger one. It turns out that, because of the intricate nature of the algebras in question, the possible positions are under rather rigid control, allowable configurations often reflecting symmetries of the quantum mechanical systems to which the algebras correspond. Symmetries, especially the hidden symmetries that subfactor theory helps to uncover, play a fundamental role in science, for they usually provide the key to reducing the complexity of the questions under study by bringing the number of free parameters that must be accounted for into a mathematically mangeable range. The search for hidden symmetries has thus become a focal point for the activities of many physical scientists. Subfactor theory provides researchers with a precise tool for understanding symmetries in a host of mathematical and physical settings. The aim of this project is to shed new light on some of the important mathematical issues that surface in this context.