On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts

TL;DR

This paper investigates the structure of mode transition and Zhu algebras for the Weyl vertex algebra at central charge 2, revealing new properties of modules and their interlocking behavior relevant to bosonic ghost modules.

Contribution

It explicitly constructs higher level Zhu algebras for the Weyl vertex algebra at c=2 and analyzes the interlocking properties of modules, advancing understanding of bosonic ghost representations.

Findings

Mode transition algebras admit strong unities.

All weak modules are induced from level-zero Zhu algebra.

Modules obtained via spectral flow are not weakly interlocked.

Abstract

We study the mode transition algebras and Zhu algebras in the setting of $\mathbb{Z}$-graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the $\beta\gamma$-system). We show that the mode transition algebras of the Weyl vertex algebra at central charge 2 admit unity elements that form a family of strong unities in the sense of Damiolini-Gibney-Krashen. The existence of unities for the mode transition algebra of the Weyl vertex algebra at central charge 2 allows us to explicitly construct all higher level Zhu algebras of the Weyl vertex algebra at central charge 2. We further analyze weak modules of the Weyl vertex algebra at central charge 2 induced from Zhu algebras, proving that every such module is already induced from the level-zero Zhu algebra. We then prove that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, and hence not strongly interlocked in the sense of Barron-Batistelli-Orosz Hunziker-Yamskulna. More generally, we show that the property of being weakly interlocked is preserved under the action of an invertible Li's $\mathbf{\Delta}$ operator. As an application, we prove that all indecomposable reducible weight modules of the Weyl vertex algebra at central charge 2 obtained via spectral flow of Zhu-induced modules are likewise not weakly interlocked. These results clarify the role of being weakly interlocked in the modularity properties of bosonic ghost modules previously studied by Ridout-Wood and Allen-Wood.