Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules

摘要

This is the first part in a series of papers in which we introduce anddevelop a natural, general tensor category theory for suitable modulecategories for a vertex (operator) algebra. This theory generalizes the tensorcategory theory for modules for a vertex operator algebra previously developedin a series of papers by the first two authors to suitable module categoriesfor a "conformal vertex algebra" or even more generally, for a "M\"obius vertexalgebra." We do not require the module categories to be semisimple, and weaccommodate modules with generalized weight spaces. As in the earlier series ofpapers, our tensor product functors depend on a complex variable, but in thepresent generality, the logarithm of the complex variable is required; thegeneral representation theory of vertex operator algebras requires logarithmicstructure. This work includes the complete proofs in the present generality andcan be read independently of the earlier series of papers. Since this is a newtheory, we present it in detail, including the necessary new foundationalmaterial. In addition, with a view toward anticipated applications, we developand present the various stages of the theory in the natural, general settingsin which the proofs hold, settings that are sometimes more general than what weneed for the main conclusions. In this paper (Part I), we give a detailedoverview of our theory, state our main results and introduce the basic objectsthat we shall study in this work. We include a brief discussion of some of therecent applications of this theory, and also a discussion of some recentliterature.