Conformal Field Theories (CFTs) appear in a variety of physical contexts, in statistical mechanics they describe the critical behavior of systems at second order phase transitions, through the AdS/CFT correspondence they are useful to the study of quantum gravity in Anti-de-Sitter spaces. Finally, they appear as fixed points in the space of Quantum Field Theories (QFTs). When describing a physical system using the language of QFT one finds a non-trivial dependence on the energy, or length scale, at which the system is probed. As we change the energy at which we probe the theory the couplings change, leading to a trajectory, or flow, in the space of QFTs. Conformal field theories, on the other hand, look the same at all length scales – they are scale invariant – and appear as fixed points in the space of QFTs. CFTs are very symmetric, apart from scale invariance, they are invariant under all conformal transformations, that is, all angle preserving transformations (see here for the effect of conformal transformations on a picture). The large number of symmetries makes these theories more tractable, while remaining physically relevant.
The goal of this project is to understand the foundations of conformal field theory. The project will start by understanding how CFTs arise in physical contexts, after which we will cover the basics of CFT. The project can then either focus on two-dimensional (where the symmetry algebra enhances) or higher-dimensional theories, and can develop in different directions corresponding to different applications of CFTs.