Topological and conformal interfaces in two-dimensional quantum field theories

边界共形场论（bCFTs）和缺陷共形场论（dCFTs）都是非常重要的主题，当我们试图将一般共形场论（CFT）技术应用于现实世界的问题时，必然会出现。当人们考虑边界或系统缺陷对CFT的影响时，它们就变得相关。这些效应降低了我们原来的CFT的对称性，这使得理论更丰富和更复杂。考虑两个二维CFT，它们在二维欧几里得空间中沿着一个界面或缺陷连接在一起。在界面上，我们可以对两个CFTs的能量动量张量施加某些条件，这定义了共形界面的概念。这些，反过来，可以分为两个子类;其中最有趣的属性是拓扑接口。例如，当把拓扑扭曲的四维N=4超杨-米尔斯理论紧化为二维时，这些界面自然出现，在那里它们被实现为四维中某些特定威尔逊线算子的图像。另一类共形界面被称为全反射界面，其中缺陷可以被视为描述具有边界的CFT，分别针对两个CFT中的每一个，这意味着两个CFT是解耦的。我们将从中得到启发，以发展新的概念和方法，可以应用到更一般的情况下，RCFTs。这个项目的目的是研究最近开发的和旧的CFT技术和方法，并将其应用于研究共形界面，从理性共形场理论（RCFTs）开始，特别是尚未在文献中探索的模型，如某些Wess-Zumino-维滕模型。开始，目标之一是看看所谓的缺陷或边界算子如何与批量算子相互作用，并找到它们的融合规则代数。此外，通过某种构造，对CFTs之间的缺陷进行分类的问题与对CFTs之间的重整化群流进行分类有关。因此，我们将继续我们的项目，找出我们的初步结果意味着两个CFTs之间的重整化群流。原则上有两种方法来描述RCTF，第一种是基于顶点代数和某些Frobenius代数的拓扑场论方法，而第二种是基于A. Belavin，A.M. Polyakov和A.B. Zamolodchikov在1984年在这个项目中，我们将尝试从“数学”和“物理”方法中汲取思想，并将它们联合收割机结合起来，以便继续解决我们的问题。最后，应该提到的是，由于CFTs位于重整化群流的端点，它们可以表征量子场论（QFT）的紫外和红外极限，因此通过研究CFTs的边界和缺陷，我们可以在QFT空间中更大的域上使用强大的CFT技术。

Boundary Conformal Field theories (bCFTs) and Defect Conformal Field Theories (dCFTs) are both immensely important subjects that necessarily emerge when we try to apply general Conformal Field Theory (CFT) techniques in real-world problems. They become relevant when one considers the effect of boundaries or system defects on a CFT. These effects reduce the symmetry of our original CFT which makes the theory richer and more complex.Consider two two-dimensional CFTs which are joined together along an interface - or defect - in 2-dimensional Euclidean space. On the interface, we can impose certain conditions for the energy-momentum tensors of the two CFTs and that defines the notion of a conformal interface. These, in turn, can be split up into two subcategories; the one with the most interesting properties is that of topological interfaces. For example, these interfaces appear naturally when compactifying topologically twisted four-dimensional N=4 super Yang-Mills theory to two dimensions, where they are realized as the images of some specific Wilson line operators in four dimensions. The other category of conformal interfaces are called totally reflective interfaces where the defect can be regarded as describing a CFT with a boundary, individually for each of the two CFTs, meaning that the two CFTs are decoupled.There are numerous recent results in the case of minimal models, such as the critical three-state Potts model, from which we will draw inspiration in order to develop new concepts and methods that can be applied to the more general case of RCFTs. This project aims to study recently developed and old CFT techniques and methods and apply them to study conformal interfaces, starting with Rational Conformal Field Theories (RCFTs) and especially to models not yet explored in the literature such as certain Wess-Zumino-Witten models. To begin with, one of the goals is to see how the so-called defect - or boundary - operators interact with bulk operators and find their fusion rule algebras. Furthermore, via a certain construction, the problem of classifying defects between CFTs is related to classifying renormalization group flows between the CFTs. Therefore, we would proceed our project with finding out what our initial results imply for the renormalization group flows between the two CFTs. There are in principle two ways to describe RCTFs, the first is the Topological Field Theory approach which is based on vertex algebras and certain Frobenius algebras, while the second is the standard Operator Product Expansion approach which is based on the seminal paper of A. Belavin, A.M. Polyakov and A.B. Zamolodchikov in 1984. In this project we will try to draw ideas from both the "mathematics" and the "physics" approach and combine them in order to proceed in our problem. Finally, it should be mentioned that because CFTs lie at the endpoints of renormalization group flows, they can characterize the ultraviolet and infrared limits of Quantum Field Theories (QFTs), hence by studying boundaries and defects for CFTs we can use powerful CFT techniques on a much larger domain in the space of QFTs.