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

TL;DR

This paper develops a unified framework for $(G, ext{chi}_ ext{phi})$-equivariant $ ext{phi}$-coordinated modules in vertex algebras, establishing new formulas, constructions, and applications to affine and Virasoro algebras.

Contribution

It introduces a generalized theory of equivariant $ ext{phi}$-coordinated modules for vertex algebras, including new formulas and constructions, and applies these to important classes like affine and Virasoro algebras.

Findings

Established a generalized commutator formula.

Constructed vertex algebras with equivariant $ ext{phi}$-coordinated modules.

Determined modules for affine and Virasoro vertex algebras.

Abstract

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.