Higher level Zhu algebras and modules for vertex operator algebras

TL;DR

This paper explores the relationship between higher level Zhu algebras and modules for vertex operator algebras, clarifying previous theorems, and illustrating how these structures influence the construction of indecomposable modules, including logarithmic examples.

Contribution

It provides corrected theorems and new insights into how higher level Zhu algebras determine the types of indecomposable modules for vertex operator algebras.

Findings

The structure of $A_1(V)$ influences the construction of indecomposable $V$-modules.

Examples with Heisenberg and Virasoro algebras illustrate the impact of $A_1(V)$ on module induction.

Constructed logarithmic modules for the Virasoro algebra that are not highest weight.

Abstract

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra $V$, we study the relationship between various types of $V$-modules and modules for the higher level Zhu algebras for $V$, denoted $A_n(V)$, for $n \in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n-1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable $V$-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are $V$-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the $V$-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when $V$ is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable $V$-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.