Vertex F-algebras and their ϕ-coordinated modules

In this paper, for every one-dimensional formal group F we formulate and study a notion of vertex F-algebra and a notion of ϕ-coordinated module for a vertex F-algebra where ϕ is what we call an associate of F. In the case that F is the additive formal group, vertex F-algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex F-algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study ϕ-coordinated modules for a general vertex Z-graded algebra V with ϕ specialized to a particular associate of the additive formal group and we give a canonical connection between V-modules and ϕ-coordinate modules for a vertex algebra obtained from V by Zhu’s change-of-variables theorem.