2D Conformal Field Theory and Liouville theory

Abstract

Conformal field theory is a formalism encountered in many branches of physics, such as String Theory and condensed matter physics. Given the wide range of its applicability it has become a subject of extensive study and research. In such field theoretic descriptions we are usually interested in computing observables called correlators. Two dimensional CFTs are important not only because they are simplified by the presence of an infinite dimensional symmetry algebra, thereby making it easier to compute correlators, but also because they play a very important role in the Polyakov String action. Liouville theory emerges when one couples a conformally invariant field to a two dimensional quantized gravitational background. The gravity sector of Liouville theory matches that of non-critical string theory, hence assigning it more importance. In this project we first try to understand the conceptual and computational aspects of two dimensional conformal field theories. Thereafter, the discussion will move onto Liouville theory, which is an example of an irrational conformal Field theory. This will include the study of how Liouville theory emerges from two-dimensional quantum gravity plus a conformal Field theory, and studying the DOZZ proposal: The conjectural formula for the three point structure constant in Liouville theory.