The structure of parafermion vertex operator algebras

Abstract

It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A (1) 1 of level k coincides with a certain W -algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.

The Structure of Parafermion Vertex Operator Algebras: General Case

“…The building block of the Kac-Moody Lie algebras is the 3-dimensional simple Lie algebra sl 2 associated to any real root. The generator results for the parafermion vertex operator algebras given in this paper and [6] show that the parafermion vertex operator algebras associated to the affine Lie algebra A (1) 1 are also the building block of general parafermion vertex operator algebras. We hope this fact will be important in the future study of the representation theory for the parafermion vertex operator algebra.…”

“…Theorem 2.1 has been obtained in [6] previously in the case g = sl 2 . The proof given here simplifies the proof of Theorem 2.1 in [6].…”

“…Then P α is isomorphic to the W -algebra W (2, 3, 4, 5) [1] by [6,Theorem 3.1] with k replaced by k α , i.e., P α is isomorphic to N (sl 2 , k α ).…”

“…Although the parafermion field theory has been studied for more than two decades, the mathematical investigation of the parafermion vertex operator algebras have been limited by a lack of understanding of the structural theory of these algebras. The goals of this paper and [4][5][6] are to alleviate this situation.…”

Construction of Holomorphic Vertex Operator Algebras of Central Charge 24 Using the Leech Lattice and Level p Lattices

“…Next we review the definition of parafermion VOAs and some of the basic properties [10,19]. Definition 3.2.…”