Formula omitted]-graded modules for [formula omitted]-graded modular vertex superalgebras

Abstract

In this paper, we study Z-graded vertex superalgebras over a general field of an odd prime characteristic. In particular, we study affine vertex superalgebras Vgˆ(ℓ,0) and a family of quotient vertex superalgebras Vgˆχ(ℓ,0) with g assumed to be a restricted Lie superalgebra. Among the main results, we obtain a necessary and sufficient condition for a gˆ-module of level ℓ to be a Vgˆχ(ℓ,0)-module. Furthermore, we study the special case with g=g1¯ and introduce a quotient vertex superalgebra Lgˆ(ℓ,0) of Vgˆ(ℓ,0). As the main results, we prove that for ℓ≠0, vertex superalgebra Lgˆ(ℓ,0) is simple, the adjoint module is the only irreducible N-graded module up to equivalence, and every Lgˆ(ℓ,0)-module is isomorphic to the direct sum of some copies of Lgˆ(ℓ,0).