The duality between vertex operator algebras and coalgebras, modules and comodules

Abstract

We construct an equivalence between the categories of vertex op- erator algebras and vertex operator coalgebras. We then investigate to what degree weak modules, generalized modules and ordinary modules carry corre- sponding comodule structures, as well as when various comodules carry module structure. In this paper we describe a universal procedure for constructing an infinite fam- ily of multiplications on the dual space of a vertex operator algebra, and then show that this procedure yields a vertex operator coalgebra. This construction provides an equivalence of categories between the category of vertex operator algebras and the category of vertex operator coalgebras. Nearly all the axioms of vertex operator coalgebras are proven directly from the corresponding axiom of a vertex operator algebra. However, in the case of the truncation condition, such a parallelism does not exist. Using equivalent characterizations of vertex operator algebras and ver- tex operator coalgebras which refer to the weights of the operators involved, we are able to maintain the parallel correspondence in our proof. This transition to an equivalent characterization may seem insignificant but has implications when we shift our attention to modules. We begin by recalling the definition of weak mod- ules, generalized modules and ordinary modules over a vertex operator algebra, then discuss the corresponding comodule notions over a vertex operator coalgebra. We then investigate whether a particular type of module structure induces a corre- sponding comodule structure. As may be suspected from the above discussion, the truncation condition plays a central role. Finally, we discuss how contragredient modules allow us to construct module and comodule structures on the same space in addition to constructing them on the dual space. The motivation for these questions dates back to the early days of the develop- ment of the theory of vertex operator algebras (and vertex algebras). Vertex oper- ator algebras arose in the 1980s in conformal field theory, with near simultaneous motivation coming from work in infinite-dimensional Lie algebras, finite sporadic