On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules

Abstract

This paper studies integral forms for affine vertex superalgebras and their modules. We first obtain integral forms for the universal enveloping superalgebra $U(\hat{\g})$ of an affine Lie superalgebra $\hat{\g}$. For $\g$ a basic classical Lie superalgebra, we give another construction of integral forms for $U(\hat{\g})$ using Chevalley basis, generalizing Kostant-Garland integral form theory. Then we apply the theory to construct integral forms for vertex (operator) superalgebras based on affine Lie superalgebras and their modules, we also investigate when an integral form contains the conformal vector, and integral forms in contragredient modules for vertex operator superalgebras.