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

Katrina Barron, Justine Fasquel, Florencia Orosz Hunziker, Gaywalee Yamskulna

TL;DR

This work extends the theory of Zhu algebras and mode transition algebras to $\mathbb{Z}$-graded vertex algebras and applies it to the Weyl (bosonic ghost) system at $c=2$. It proves that the $d$-th mode transition algebras $\mathfrak{A}_d$ admit unity, yielding a Splitting Theorem that expresses higher level Zhu algebras as a direct sum, and it establishes a strong-unity structure for these algebras. Specializing to the Weyl vertex algebra $\mathcal{V}$, the authors compute $\mathfrak{A}_d(\mathcal{V}) \cong \mathscr{W} \otimes \mathrm{Mat}_{|P_2(d)|}(\mathbb{C})$ and deduce explicit descriptions of $\mathrm{Zhu}_d(\mathcal{V})$, revealing a rich, block-matrix-like higher-level structure. The paper then develops a Zhu-induced module theory for $\mathbb{Z}_{\ge 0}$-gradable modules, studies Li's $\mathbf{\Delta}$-twists and spectral flow, and proves that indecomposable reducible weight $\mathcal{V}$-modules arising from Zhu induction or spectral flow are not weakly interlocked, with implications for modularity and graded traces in bosonic ghost theories.

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 $βγ$-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 $\mathbfΔ$ 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.

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

TL;DR

This work extends the theory of Zhu algebras and mode transition algebras to -graded vertex algebras and applies it to the Weyl (bosonic ghost) system at . It proves that the -th mode transition algebras admit unity, yielding a Splitting Theorem that expresses higher level Zhu algebras as a direct sum, and it establishes a strong-unity structure for these algebras. Specializing to the Weyl vertex algebra , the authors compute and deduce explicit descriptions of , revealing a rich, block-matrix-like higher-level structure. The paper then develops a Zhu-induced module theory for -gradable modules, studies Li's -twists and spectral flow, and proves that indecomposable reducible weight -modules arising from Zhu induction or spectral flow are not weakly interlocked, with implications for modularity and graded traces in bosonic ghost theories.

Abstract

We study the mode transition algebras and Zhu algebras in the setting of -graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the -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 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.
Paper Structure (23 sections, 21 theorems, 129 equations)

This paper contains 23 sections, 21 theorems, 129 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Organization and Main Results
  3. Acknowledgements
  4. Preliminaries
  5. Higher level Zhu algebras and mode transition algebras for Z-graded vertex algebras
  6. Zhu algebras associated to a Z-graded vertex algebra
  7. The Lie algebra of a Z-graded vertex algebra, filtrations, universal enveloping algebras, and completions
  8. The left, right, and finite universal enveloping algebras of a Z-graded vertex algebra
  9. Zhu algebras as quotients of the universal enveloping algebra
  10. The mode transition algebra for a Z-graded vertex algebra
  11. Mode transition algebras with unities and the Splitting Theorem for Zhu algebras
  12. Families of strong unities for mode transition algebras
  13. The mode transition algebra and higher level Zhu algebras for the Weyl vertex algebra at central charge 2.
  14. The Weyl vertex algebra
  15. Mode transition algebras and higher level Zhu algebras for the Weyl vertex algebra
  16. ...and 8 more sections

Key Result

Proposition 2.2

If $V$ is a $\mathbb{Z}$-graded vertex algebra, then for $n \in \mathbb{Z}_{\geq 0}$, the space $O_n(V)$ is a two-sided ideal of $V$ with respect to $*_n$, and the quotient $\mathrm{Zhu}_{n}(V) = V/O_n(V)$ is a unital associative algebra with respect to $*_n$ and ${\bf 1}$, called the level $n$ Zhu

Theorems & Definitions (62)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Proposition 2.2: ZhuDLMKDS
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 52 more