ABSTRACT
Conformal field theory has had a major impact on various aspects of modern mathematics, in particular, the theory of vertex operator algebras and Borcherds algebras, finite groups and number theory. The process of reconstructing the space of states from the vacuum expectation values of fields is familiar from axiomatic quantum field theory. In the usual Osterwalder-Schrader framework of Euclidean quantum field theory, the reflection positivity axiom guarantees that the resulting space of states has the structure of a Hilbert space. The decomposition of the space of states in terms of representations of the two vertex operator algebras throws considerable light on the problem of whether the theory is well-defined on higher Riemann surfaces. The description of representations in terms of collections of densities has a large redundancy in that many different collections of densities define the same representation. Y. Zhu’s algebra plays a central rôle in characterising the structure of a conformal field theory.
