Axiomatic conformal field theory

A new rigourous approach to conformal field theory is presented. The basic objects an families of complex-valued amplitudes, which define a meromorphi c conformal field theory (or chiral algebra) and which lead naturally to th e definition of topological vector spaces, between which vertex operators a ct as continuous operators. In fact, in order to develop the theory, Mobius invariance rather than full conformal invariance is required but it is sho wn that every Mobius theory can be extended to a conformal theory by the co nstruction of a Virasoro field. In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic prope rties. It is shown that these amplitudes can also be derived from a suitabl e collection of states in the meromorphic theory. Zhu's algebra then appear s naturally as the algebra of conditions which states defining highest weig ht representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is e xplained.