Integral forms for certain classes of vertex operator algebras and their modules

Abstract

OF THE DISSERTATION Integral Forms for Certain Classes of Vertex Operator Algebras and Their Modules by Robert H. McRae Dissertation Director: James Lepowsky We study integral forms in vertex operator algebras over C. We prove general results on when a multiple of the standard conformal vector ω can be added to an integral form of a vertex operator algebra and when intertwining operators among modules for a vertex operator algebra respect integral forms in the modules. We also show when the Z-dual of an integral form in a module for a vertex operator algebra is an integral form in the contragredient module. As examples, we consider vertex operator algebras based on affine Lie algebras and even lattices, and tensor powers of the Virasoro vertex operator algebra L( 2 , 0). In particular, we demonstrate vertex algebraic generators of integral forms for standard modules for affine Lie algebras that were first constructed in work of Garland; we reprove Borcherds’ construction of integral forms in lattice conformal vertex algebras using generators; and we find generating sets that include ω for integral forms in tensor powers L( 2 , 0)⊗n when n ∈ 4Z. We also construct integral forms in modules for these vertex operator algebras using generating sets, and we show when intertwining operators among modules for these vertex operator algebras respect integral forms in the modules.