Nonrational Conformal Field Theory

Abstract

We introduce a formalism for the construction of correlation functions in certain classes of nonrational conformal field theories. An important role is played by non-degenerate hermitian forms on the spaces of conformal blocks, which allow us to gain some control on the issues coming from the infinite-dimensionality of these spaces. Appropriate generalizations of the concept of a modular functor and of the Friedan-Shenker modular geometry are presented. It is argued that the hermitian form on the spaces of conformal blocks is in fact a scalar product when the representations involved are all unitary, which is illustrated by the case of the c>1 Virasoro conformal blocks.