Vertex $F$-algebras and their $\varphi$-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 $\varphi$-coordinated module for a vertex $F$-algebra where $\varphi$ 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 $\varphi$-coordinated modules for a general vertex $\mathbb{Z}$-graded algebra $V$ with $\varphi$ specialized to a particular associate of the additive formal group and we give a canonical connection between $V$-modules and $\varphi$-coordinate modules for a vertex algebra obtained from $V$ by Zhu’s change-of-variables theorem.