Contemporary Mathematics

Volume 343, 2004

Regularity of certain vertex operator superalgebras

Drazen Adamovic

ABSTRACT.

We present our results on representation theory of certain vertex

operator superalgebras. In particular, we consider vertex operator superal-

gebras associated to the minimal models for the Neveu-Schwarz algebra and

the N = 2 superconformal algebra. We present the results on the classifica-

tion of irreducible representation, regularity, rationality and fusion rules for

these vertex operator superalgebras. The connections with the theory of affine

Kac-Moody and Virasoro vertex operator algebras will be also discussed.

1.

Introduction

The main problems in the theory of vertex operator (super )algebras are related

to the constructions and classification of rational vertex operator (super)algebras.

The rationality of certain well-known examples of vertex operator (super)algebras

was proved in papers

[D], [DL], [Lil], [FZ], [Wn], [Al], [A2], [A3].

In

[DLM]

was introduced the notion of regular vertex operator algebra, i.e.,

rational vertex operator algebra with the property that every weak module is com-

pletely reducible. So every regular vertex operator algebra has finitely many ir-

reducible modules, and every

V

~module

is completely reducible. Further devel-

opment in theory of regular vertex operator algebras was made by

H.

Li in

[Li2]

by proving that every regular vertex operator algebra satisfies Zhu's

C

2

~finiteness

condition and that the fusion rules for irreducible modules are finite. Zhu's

C 2 ~

finiteness condition is important for studying modular invariance of characters of

vertex operator algebras (see

[Z]).

The paper

[DLM]

also gave the proof of regularity of lattice vertex opera-

tor algebras, vertex operator algebras associated to integrable representations of

affine Kac-Moody Lie algebras, and vertex operator algebras associated to minimal

models for the Virasoro algebra.

2000

Mathematics Subject Classification.

Primary 17B69, Secondary 17B67, 17B68, 81R10.

Key words and phrases.

Vertex operator superalgebras, rationality, regularity, fusion rules,

representation theory, Neveu-Schwarz algebra, N=2 superconformal algebra.

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/343/06181