Representation and category theoretic aspects of logarithmic conformal field theories

对数共形场论的表示和范畴理论方面

• 批准号: 219517345
• 负责人: Professor Dr. Christoph Schweigert
• 金额: --
• 依托单位: Bereich Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/219517345?language=en
• 关键词: Representation; category; theoretic; aspects; logarithmic

Representations of symmetry structures in physics are, in large classes of examples, completely reducible. Logarithmic conformal _eld theories are a class of quantum _eld theories with numerous applications in statistical mechanics and solid state physics; in these theories indecomposable but not irreducible representations arise.Some relevant representation categories are mathematically well-understood, e.g. via quantized universal envelopping algebras. We propose on the one hand side to make new classes of examples explicitly accessible. On the other hand, we propose to construct in general and in classes of examples physically relevant quantities using tools from representation theory and category theory.We have three detailed goals for dissertation projects:1. Construction of invariants of actions of mapping class groups which are candidates for physical correlators.2. Calculations of these invariants in concrete examples, e.g. representation categories of Nichols algebras or group algebras in \bad" characteristic.3. Description of physically relevant quantities like boundary conditions or defect types in terms of categorical quantities.

物理学中对称结构的表示在大类的例子中是完全可约的。对数共形场理论是量子场理论的一个分支，在统计力学和固体物理中有着广泛的应用。在这些理论中，出现了不可分解的但不是不可约的表示。一些相关的表示范畴在数学上是很好理解的，例如通过量子化泛代数。一方面，我们建议使新的类的例子显式访问。另一方面，我们建议使用表示论和范畴论的工具来构造一般的和实例类的物理相关量。我们有三个具体的论文项目目标：1.构造映射类群的动作不变量，作为物理映射的候选.这些不变量的计算在具体的例子中，例如表示类别的尼科尔斯代数或群代数在“坏”的特征.用分类量描述物理相关量，如边界条件或缺陷类型。