Topological and conformal interfaces in two-dimensional quantum field theories

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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.