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Rational chiral conformal field theories are organized according to theirgenus, which consists of a modular tensor category $\mathcal{C}$ and a centralcharge $c$. A long-term goal is to classify unitary rational conformal fieldtheories based on a classification of unitary modular tensor categories. Weconjecture that for any unitary modular tensor category $\mathcal{C}$, thereexists a unitary chiral conformal field theory $V$ so that its modular tensorcategory $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate amathematical program in and around this conjecture. We define a class ofextremal vertex operator algebras with minimal conformal dimensions as large aspossible for their central charge, and non-trivial representation theory. Weshow that there are finitely many different characters of extremal vertexoperator algebras V possessing at most three different irreducible modules.Moreover, we list all of the possible characters for such vertex operatoralgebras with $c$ at most 48.

On classification of extremal non-holomorphic conformal field theories