Topological and conformal interfaces in two-dimensional quantum field theories

二维量子场论中的拓扑和共形界面

• 批准号: 2616598
• 金额: --
• 依托单位: Heriot-Watt University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2616598
• 关键词: Topological; conformal; interfaces; two; dimensional

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.

边界共形场理论（BCFTS）和缺陷共形场理论（DCFTS）都是非常重要的主题，当我们尝试在现实世界中应用一般的保形场理论（CFT）技术时，必然会出现。当人们认为边界或系统缺陷对CFT的影响时，它们就变得很重要。这些效果降低了我们原始CFT的对称性，这使理论变得更丰富，更复杂。考虑两个二维CFT，它们沿二维欧几里得空间沿界面或缺陷沿界面或缺陷连接在一起。在界面上，我们可以为两个CFT的能量量张量施加某些条件，并定义了共形界面的概念。反过来，这些可以分为两个子类别。最有趣的属性是拓扑界面。例如，这些接口在压实拓扑扭曲的四维n = 4 Super Yang-Mills理论时自然而然地出现，并在两个维度上将它们实现为四个维度的某些特定的Wilson Line Operators的图像。其他类别的形式界面被称为完全反思性的界面，在该界面中可以被视为描述具有边界的CFT，单独地是两个CFT，这意味着两个CFT是脱钩的。在最小的模型中，有更多的模型，例如，我们将开发出更多的概念，我们将在最小的模型中绘制一些概念，并在这些模型中绘制新的概念，并在该模型中绘制新的概念，并在该模型中绘制序列，并在该概念中绘制了新的概念。 rcfts。该项目旨在研究最近开发的和旧的CFT技术和方法，并将它们应用于研究形式界面，从理性的保形场理论（RCFT）开始，尤其是在文献中尚未探索的模型（例如某些Wess-Zumino-witten模型）中尚未探索的模型。首先，目标之一是查看所谓的缺陷或边界操作员如何与散装操作员交互并找到其融合规则代数。此外，通过一定的结构，对CFT之间的缺陷进行分类的问题与对CFT之间的重归化组流进行分类有关。因此，我们将继续进行项目，以找出我们最初的结果暗示着两个CFT之间的重新规范化组流动。原则上有两种描述RCTF的方法，第一种是基于顶点代数和某些frobenius代数的拓扑场理论方法，而第二个是基于A.M. A. Belavin的开创性论文的标准操作员产品扩展方法。 Polyakov和A.B. Zamolodchikov于1984年。在这个项目中，我们将尝试从“数学”和“物理学”方法中汲取思想，并结合起来以解决我们的问题。最后，应该提到的是，由于CFTS位于重新归一化组流的终点，它们可以表征量子场理论（QFTS）的紫外线和红外限制，因此，通过研究CFT的边界和缺陷，我们可以在QFTS空间中使用更大的较大的CFT技术来使用CFT的界限和缺陷。