Classification and $C_2$-cofiniteness of rational vertex operator algebras generated in low conformal weights

Abstract

We obtain classification results for rational vertex operator algebras generated in low conformal weights. More precisely, we prove that every rational vertex operator algebra of OZ type strongly generated by its conformal weight two subspace $V_2$ is isomorphic to a finite tensor product of Virasoro minimal models. We also prove that every rational vertex operator algebra of CFT type generated as a vertex algebra by its conformal weight one subspace $V_1$ is isomorphic to a finite tensor product of simple affine vertex operator algebras associated with finite-dimensional simple Lie algebras at positive integral levels. Consequently, Zhu's conjecture that rationality implies $C_2$-cofiniteness holds for both classes, and every such rational vertex operator algebra is regular.