$(G,χ_ϕ)$-equivariant $ϕ$-coordinated modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for vertex algebras, where $G$ is a group with $χ_ϕ$ a linear character of $G$ and $ϕ$ is an associate of the one-dimensional additive formal group. The theory of $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for nonlocal vertex algebra is established in \cite{JKLT}. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules. Furthermore, for any conformal algebra $\mathcal{C}$, we construct a class of Lie algebras $\widehat{\mathcal{C}}_ϕ[G]$ and prove that restricted $\widehat{\mathcal{C}}_ϕ[G]$-modules are exactly $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for the universal enveloping vertex algebra of $\mathcal{C}$. As an application, we determine the $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for affine and Virasoro vertex algebras.