Volume 343, 2004
Regularity of certain vertex operator superalgebras
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.
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].
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
is completely reducible. Further devel-
opment in theory of regular vertex operator algebras was made by
by proving that every regular vertex operator algebra satisfies Zhu's
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
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.
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