Representation theoretic tools for equivariant and orbifold conformal field theories

有理二维共形量子场论可以用模张量范畴（如仿射Kac-Moody代数的表示范畴的子范畴）中的可分对称Frobenius代数及其表示理论来描述；因此，它们特别适合用数学方法来处理。TFT结构为这些理论的一组一致的相关器的存在提供了建设性的证明。在本项目中，将研究该理论的g等变版本。等变版本承认，在适当的条件下，一个轨道结构以类似于其他数学理论中的“传递到商”的方式产生了非等变理论的新例子。该项目有三个主要目标：1。g等变模张量范畴实例的构造。通常，这样的构造分两步完成：首先，构造并赋予一组模块类别，在第二步中，通过附加结构完成等变数据。置换群对同模张量范畴的张量积的作用提供了一个特别容易处理的例子。利用基于仿射Kac-Moody代数的扭曲表示的表示范畴，给出了一个李论例子，以及相应的紧连李群、连通李群和单连通李群的自同构群对它们的作用。2. 在三范畴情境下，利用伪单轨及其表征理论对轨道结构的概念性理解。一个特别有趣的例子是Brauer三人组。3. RCFT相关器TFT构造的等变推广。

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.