On classification of extremal non-holomorphic conformal field theories

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $C$ and a central charge c. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $C$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $C_V$ is $C$. In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras $V$ possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with $c⩽48$. (paper)