On C2-cofiniteness of parafermion vertex operator algebras

It is proved that any locally regular vertex operator algebra is C-2-cofinite and the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac-Moody algebra A(1)((1)) implies the C-2-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 C-2-cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determined explicitly. (C) 2010 Elsevier Inc. All rights reserved.