Twisted φ-coordinated modules for nonlocal vertex algebras

In this paper, we study what we call twisted phi-coordinated modules for nonlocal vertex algebras including weak quantum vertex algebras and we give a conceptual construction of nonlocal vertex algebras and their twisted phi-coordinated modules. As an illustrating example, we study certain vertex algebras and their twisted phi-coordinated modules, associated to certain infinite dimensional Heisenberg Lie algebras. As a reformulation and a slight extension of a result of Lepowsky, and a result of Doyon, Lepowsky and Milas, we also give a canonical connection between twisted modules for a general 1/T-Z-graded nonlocal vertex algebra V and twisted phi-coordinated modules for the nonlocal vertex algebra obtained from V by Zhu's change-of-variable theorem. (C) 2019 Elsevier Inc. All rights reserved.