Fusion Algebras and Differential Equation Approach to Rational Conformal Field Theories

Constraints on conformal dimensions and central charge of Rational Conformal Field Theories (RCFTs) coming from analytic closure of 4-point functions on the sphere and modular invariance of partition function on the torus are reviewed. These constraints, together with modular invariance ones and a deeper understanding of the structure of Fusion Algebras (FAs) can help a lot in the classification of RCFTs. In particular we show how FAs with 2 and 3 primary operators can be completely classified.