Nonunitary Conformal Field Theories with Boundary

Abstract

In this thesis, we review and report a few result on the conformal field theories with boundary. We start with a review of the theory. Considering applications in wide areas of theoretical physics, from the condensed matter physics to the string theory which unifies the gravity with quantum field theory, we explain the minimal models, Wess-Zumino-Witten models and also review coset construction. We define these theories with the central charges, the conformal dimensions and the operator product expansions. Using these, we show how to compute the correlation functions. In addition, we consider the geometry of a torus to compute the modular invariant partition functions. We review Verlinde's formula which relates the fusion rules to the modular properties of the partition functions. These aspects are important since they are related to the previous quantities as well as their applications th the statistical mechanics and the string theory. Then, we concentrate on the boundary conformal field theories which are recently studied actively due to applications in ordinary space-time like the quantum impurity phenomena known as the Kondo problem. Here, we start reviewing Cardy's approach who started this field. The boundary partition functions are intimately related to the modular transformation properties and often determines the degeneracies and conformally invariant boundary conditions completely. Then, we explain our research on the non-unitary conformal field theories which are very interesting since they are applied to the condensed matter like quantum Hall effect and also to the quantum gravity. In particular, we report our new result on the boundary effect in the non-unitary models in general and show explicit computations for the simplest models, the Yang-Lee edge singularity model.