Axiomatic Conformal Field Theory

TL;DR

This paper introduces a rigorous, axiomatic framework for conformal field theory using complex amplitudes, topological vector spaces, and vertex operators, extending M"obius invariance to full conformal invariance via Virasoro fields.

Contribution

It develops a new axiomatic approach to conformal field theory based on meromorphic amplitudes and topological vector spaces, connecting Zhu's algebra to representation classification.

Findings

Amplitudes define a meromorphic conformal field theory.

Every M"obius theory extends to a conformal theory with a Virasoro field.

Zhu's algebra characterizes highest weight representations.

Abstract

A new rigorous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, M\"obius invariance rather than full conformal invariance is required but it is shown that every M\"obius theory can be extended to a conformal theory by the construction 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 properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained.