Modular Symmetry in Conformal Field Theory

Abstract

The classification of conformal field theories is an important ongoing research area with applications ranging from high-energy physics to condensed-matter systems. Modular invariance has been an invaluable tool in the study of two-dimensional conformal field theories. We review a method of classification using modular-invariant linear differential equations which is based on two parameters: n, the number of characters and l, the number of zeroes of the Wronskian of the differential equation. Previously, this method has been successful for theories with a small number of characters and when l < 6. We provide new results giving a simple and complete construction of consistent solutions for all values of l>= 6 in the case of two-character theories. We further illustrate our method in the specific case of l= 6 where we realise some new theories as novel cosets of meromorphic conformal field theories.