Project IV 2022-23

Introduction to Conformal Field Theory

Madalena Lemos (email)

Description

special conformal transformation.gif Symmetry plays an important role in physics -- symmetrical systems are simpler to describe, the larger the symmetries the more constrained and tractable the system. Symmetries would be of no use, however, if the symmetrical systems we can describe had no relation to the real physical systems we care about. Fortunately this is not the case, and symmetries have been our guide in solving physical systems, going back to the work of E. Noether, either as exact symmetries, or as approximate symmetries, taken as a starting point to solve less-symmetric cases. Most physical systems depend on the scale at which they are probed -- they look and behave very differently at different scales -- scale invariant theories, on the other hand, do not. Most scale invariant systems enjoy a larger symmetry called conformal symmetry (i.e., invariance under all angle-preserving transformations - an example is shown on the right, and see here for the effect of conformal transformations on a picture). Despite the very symmetric nature of these theories they remain very much relevant to physics -- such theories are called Conformal Field Theories (CFTs). 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, being scale invariant, appear at the end-points of these flows, and thus play a prominent role in the study of the landscape of QFTs.

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, starting with understanding their place in the space of Quantum Field Theories, 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.

Pre-requisites

Co-requisites

If you are unsure about the pre- and co-requisites please feel free to get in touch.

References