On C2-cofiniteness of parafermion vertex operator algebras

It is proved that any locally regular vertex operator algebra is C2-cofinite and the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac–Moody algebra implies the C2-cofiniteness of parafermion vertex operator algebras associated to integrable highest weight modules for any affine Kac–Moody algebra. In particular, the parafermion vertex operator algebra associated to an integrable highest weight module of small level for any affine Kac–Moody algebra is C2-cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determined explicitly.