Logarithmic tensor product theory for generalized modules for a conformal vertex algebra, Part I

TL;DR

This paper extends tensor product theory for modules over conformal vertex algebras to include non-semisimple modules with generalized weight spaces, involving complex variables and logarithms, laying groundwork for braided tensor categories.

Contribution

It develops a logarithmic tensor product theory for conformal vertex algebra modules, accommodating non-semisimple modules and logarithmic intertwining operators, expanding previous semisimple-focused frameworks.

Findings

Introduces logarithmic intertwining operators for generalized modules.

Establishes tensor product functors involving complex variables and logarithms.

Prepares for construction of braided tensor category structures in subsequent work.

Abstract

We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ``conformal vertex algebra'' or even more generally, for a "M\"obius vertex algebra.'' We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is involved. This first part is devoted to the study of logarithmic intertwining operators and their role in the construction of the tensor product functors. Part II of this work will be devoted to the construction of the appropriate natural associativity isomorphisms between triple tensor product functors, to the proof of their fundamental properties, and to the construction of the resulting braided tensor category structure. This work includes the complete proofs in the present generality and can be read independently of the earlier series of papers.