Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{ C}$ 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 $\mathcal{ C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{ C}_V$ is $\mathcal{ C}$. 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 $c$ at most 48.
Categories: math-ph math.MP math.QA