Finite growth representations of infinite Lie conformal algebras

We classify all finite growth representations of all infinite rank subalgebras of the Lie conformal algebra gc_1 that contain a Virasoro subalgebra.Comment: 22 page

Integrable highest weight modules over affine superalgebras and Appell's function

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level~1 case. The analysis of this construction shows, in particular, that in the simplest case of the $s\ell (2|1)$ level~1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras.Comment: 54 page

Vertex Operator Superalgebras and Their Representations

After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We use the machinery established above to find the rationality conditions, classify the representations and compute the fusion rules. In the appendix, we present explicit formulas for singular vectors and defining relations for the integrable representations of super affine algebras. These formulas are not only crucial for the theory of the corresponding vertex operator superalgebras and their representations, but also of independent interest.Comment: 50 pages, to appear in Contemporary Mathematic