Representation theoretic tools for equivariant and orbifold conformal field theories

等变和环折共形场论的表示理论工具

• 批准号: 122827613
• 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: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/122827613?language=en
• 关键词: Representation; theoretic; tools; equivariant; orbifold

Rational two-dimensional conformal quantum field theories can be described by separable symmetric Frobenius algebras in modular tensor categories – e.g. subcategories of representation categories of affine Kac-Moody algebras – and their representation theory; they are thus particularly amenable to a mathematical treatment. The TFT construction furnishes a constructive proof of existence for a consistent set of correlators for these theories. In the present project, a G-equivariant version of the theory shall be studied. The equivariant version admits, under suitable conditions, an orbifold construction yielding – in a similar way as “passing to the quotient” in other mathematical theories – new examples of non-equivariant theories. The project has three major aims: 1. The construction of examples of G-equivariant modular tensor categories. Typically, such a construction is done in two steps: first, a family of module categories is constructed and endowed, in a second step, by additional structure completing the equivariant data. A particularly tractable example is provided by the action of the permutation group on the tensor product of identical modular tensor categories. A Lie-theoretic example is provided by representation categories based on twisted representations of affine Kac-Moody algebras and the action of the automorphism group of the corresponding compact, connected and simply-connected Lie group on it. 2. A conceptual understanding of the orbifold construction using pseudomonads and their representation theory in a three-categorical setting. A particularly interesting example should be provided by the Brauer three-group. 3. An equivariant generalization of the TFT construction of RCFT correlators.

有理二维共形量子场论可以用模张量范畴（例如仿射卡茨-穆迪代数的表示范畴的子范畴）中的可分对称弗罗贝纽斯代数及其表示理论来描述;因此它们特别适合于数学处理。TFT结构为这些理论提供了一组一致的证明。在本项目中，将研究该理论的G-等变版本。等变的版本在适当的条件下允许一个轨道结构产生--类似于其他数学理论中的“通到商”--非等变理论的新例子。该项目有三个主要目标：1。 G-等变模张量范畴的例子的构造。通常，这样的构造分两步完成：第一步，构造一个模块范畴族，并在第二步中，通过附加结构完成等变数据。一个特别容易处理的例子是置换群对相同模张量范畴的张量积的作用。通过仿射Kac-Moody代数的扭表示范畴以及相应的紧、连通和单连通李群的自同构群在其上的作用，给出了李群理论的一个例子。 使用假单胞菌及其在三范畴环境中的表征理论对眶状结构的概念性理解。一个特别有趣的例子应该是布劳尔三群。3. RCFT变换器TFT结构的等变推广。