Regularity of Rational Vertex Operator Algebras

✍ Scribed by Chongying Dong; Haisheng Li; Geoffrey Mason

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 313 KB
• Volume: 132
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1681

✦ Synopsis

Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ and MS]), single out an important class of vertex operator algebras. Most vertex operator algebras which have been studied so far are rational vertex operator algebras. Familiar examples include the moonshine module V < [B, FLM, D2], the vertex operator algebras V L associated with positive definite even lattices L [B, FLM, D1], the vertex operator algebras L(l, 0) associated with integrable representations of affine Lie algebras [FZ] and the vertex operator algebras L(c p, q , 0) associated with irreducible highest weight representations for the discrete series of the Virasoro algebra [DMZ and W].

A rational vertex operator algebra as studied in this paper is a vertex operator algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex operator algebra with this property is called a regular vertex operator algebra. One motivation for studying such vertex operator algebras is trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex operator algebras L(l, 0) for certain rational l (cf. [KS and MW]).

In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the rational vertex operator algebras V < , L(l, 0) for positive integrals l, L(c p, q , 0) and V L for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules.