Modular geometry and the classification of rational conformal field theories

Abstract

I review a recently. developed procedure to classify all conformal field theories with a finite number of characters. This method involves writing the most general modular-invariant differential equation on the moduli space of the torus, and looking for solutions which satisfy the axioms of conformal field theory. On identifying these solutions with the genus-I characters, one can then reconstruct the primary field con tent, the fusion rules, the correlation functions and the chiral algebra of the associated theory. Contour-integral representations of Feigin-Fuchs type are proposed for the characters.