Rational Conformal Field Thoeries: Classification and Applications

Abstract

Rational Conformal Field Theories (RCFTs) are used to describe many physical systems including, but not limited to, the Fractional Quantum Hall effect. In that effect, classifying RCFTs is an interesting and tractable problem. In this thesis we first understand and develop the formalism of CFTs and specialise to WZW models to study examples of RCFTs. Using the techniques of Modular Invariance, we use the Modular Linear Differential Equations (MLDE) approach in our attempt of classification of RCFT. We note that the contour integral representation of certain characters lends itself very well to understand the monodromy properties of characters. This leads us to developing an algorithm to compute the modular S-matrix for arbitrary number of characters, without specifying any chiral algebra. Subsequently we explore the space of solutions of the order 3 MLDE, and discover infinite families of solutions known as quasi-characters. Quasi-characters are solutions of the MLDE with integer but not necessarily positive q-expansion. The novel coset construction is used extensively to construct the families of solution. Using these quasi-characters we can construct new admissible characters. Following these original developments, we explore the application of RCFT in the Fractional Quantum Hall effect, after covering the basics of the Hall effect.