$(G,\chi_\phi)$-equivariant $\phi$-coordinated modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for vertex algebras, where $G$ is a group with $\chi_\phi$ a linear character of $G$ and $\phi$ is an associate of the one-dimensional additive formal group. The theory of $(G,\chi_\phi)$-equivariant $\phi$-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 $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules. Furthermore, for any conformal algebra $\mathcal{C}$, we construct a class of Lie algebras $\widehat{\mathcal{C}}_\phi[G]$ and prove that restricted $\widehat{\mathcal{C}}_\phi[G]$-modules are exactly $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for the universal enveloping vertex algebra of $\mathcal{C}$. As an application, we determine the $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for affine and Virasoro vertex algebras.