The level two Zhu algebra for the Heisenberg vertex operator algebra

TL;DR

This paper explicitly computes the level two Zhu algebra for the Heisenberg vertex operator algebra using minimal information, revealing its noncommutative nature at level two and proposing a conjecture for higher levels.

Contribution

The authors determine the level two Zhu algebra for the Heisenberg VOA with minimal data, extending understanding of its algebraic structure and noncommutativity at this level.

Findings

Level two Zhu algebra for Heisenberg VOA is noncommutative.

Method uses minimal information and previous general results.

Conjecture proposed for the structure at levels higher than two.

Abstract

We determine the level two Zhu algebra for the Heisenberg vertex operator algebra $V$ for any choice of conformal element. We do this using only the following information for $V$: the internal structure of $V$; the level one Zhu algebra of $V$ already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for $V$ with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in "On generators and relations for higher level Zhu algebras and applications", by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level $n$ Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level $n=2$. We also give a conjecture for the structure of the level $n$ Zhu algebra for the Heisenberg vertex operator algebra, for any $n >2$.