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Vertex algebras associated to abelian current algebras

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.