Vertex algebras associated to abelian current algebras

TL;DR

This paper constructs a family of vertex algebras linked to abelian current algebras, demonstrating their quasi-conformal, strongly graded structure and verifying key properties for logarithmic tensor category theory.

Contribution

It introduces a new class of vertex algebras associated with abelian current algebras and establishes their foundational properties for logarithmic module theory.

Findings

Vertex algebras are quasi-conformal and strongly $ $-graded.

Modules and logarithmic modules are constructed and analyzed.

Convergence and extension properties are verified for the tensor category framework.

Abstract

We construct a family of vertex algebras associated to the current algebra of finite-dimensional abelian Lie algebras along with their modules and logarithmic modules. We show this family of vertex algebras and their modules are quasi-conformal and strongly $\N$-graded and verify convergence and extension property needed in the logarithmic tensor category theory for strongly graded logarithmic modules developed by Huang, Lepowsky and Zhang.