Introduction

Belavin-Polyakov-Zamodolochikov ([BPZ]) initiated conformal field theory as a certain limit of the theory of the two-dimensional lattice model. This theory has a deep relationship with string theory and a rich mathematical structure. It is a two-dimensional quantum field theory invariant under conformal transformations; in fact, as we shall see below, it is invariant under a much bigger group of transformations, and this gives a relationship with the moduli space of algebraic curves ([FS], [EO]).

A typical example of conformal field theory is abelian conformal field theory, the theory of free fermions over a compact Riemann surface. For a mathematically rigorous treatment of abelian conformal field theory we refer the reader to [KNTY]. This theory has a deep relationship with various fields of mathematics, such as the moduli theory of algebraic curves, KP hierarchy, theta functions, complex cobordism rings and formal groups ([KNTY], [KSU2], [KSU3]).

For non-abelian conformal field theory the first mathematically rigorous treatment was given by Tsuchiya-Kanie ([TK]), who constructed the theory over ℙ1. Later Tsuchiya-Ueno-Yamada ([TUY]) generalized this to algebraic curves of arbitrary genus.

Let us explain briefly the main ideas of conformal field theory. It can be decomposed into two parts, holomorphic and anti-holomorphic, and in the following we shall only consider the holomorphic theory. This is often called chiral conformal field theory by the physicists.