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Balanced root systems and a Schellekens-type list for holomorphic vertex operator algebras of central charge $32$

Maneesha Ampagouni, Geoffrey Mason, Michael H. Mertens

TL;DR

This work introduces balanced root systems for holomorphic VOAs with central charges $c=32$ and $c=40$, extending Niemeier/Schellekens-type data to include levels and supplemented root systems. It proves that for these charges, the VOA’s Virasoro vector aligns with the subVOA’s Virasoro structure, enabling a Schellekens-type classification where the root data satisfy a delicate balance condition. The authors derive explicit candidate root-system lists, analyze the realizability via lattice theories and DGM orbifolds, and employ modular and Jacobi-form constraints alongside computational tests to prune candidates. The combined theoretical and computational approach narrows the landscape to a finite set of possible structures, providing concrete classifications and laying groundwork for further exploration of higher central charges in holomorphic VOAs.

Abstract

We study a special class of holomorphic vertex operator algebras (VOAs) that we call \emph{balanced}.\ For a balanced, holomorphic VOA $V=\mathbb{C}\mathbf{1}\oplus V_1\oplus\dots$ with $c=32$ or $40$ we show that the Virasoro vectors of $V$ and the subVOA generated by $V_1$ coincide and use this result to provide a Schellekens-type list of possible root systems that may occur.

Balanced root systems and a Schellekens-type list for holomorphic vertex operator algebras of central charge $32$

TL;DR

This work introduces balanced root systems for holomorphic VOAs with central charges and , extending Niemeier/Schellekens-type data to include levels and supplemented root systems. It proves that for these charges, the VOA’s Virasoro vector aligns with the subVOA’s Virasoro structure, enabling a Schellekens-type classification where the root data satisfy a delicate balance condition. The authors derive explicit candidate root-system lists, analyze the realizability via lattice theories and DGM orbifolds, and employ modular and Jacobi-form constraints alongside computational tests to prune candidates. The combined theoretical and computational approach narrows the landscape to a finite set of possible structures, providing concrete classifications and laying groundwork for further exploration of higher central charges in holomorphic VOAs.

Abstract

We study a special class of holomorphic vertex operator algebras (VOAs) that we call \emph{balanced}.\ For a balanced, holomorphic VOA with or we show that the Virasoro vectors of and the subVOA generated by coincide and use this result to provide a Schellekens-type list of possible root systems that may occur.
Paper Structure (25 sections, 21 theorems, 102 equations, 4 tables)

This paper contains 25 sections, 21 theorems, 102 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Balanced root systems
  3. Root systems and Lie algebras
  4. Balanced root systems
  5. Enumerating balanced root systems
  6. Strongly rational and holomorphic VOAs
  7. Modular forms for $\mathrm{SL}_2(\mathbf{Z})$
  8. Trace functions and the character of $V$
  9. The supplemented root system
  10. Some Virasoro vectors
  11. Balanced VOAs
  12. The cases $c=32$ and $40$
  13. The abelian case
  14. Holomorphic VOAs of pure power type
  15. DGM-orbifolds
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose that $V=\mathbf C\mathbf{1}\oplus V_1\oplus...$ is a balanced, holomorphic VOA with $c=32$ such that $V_1$ is a semisimple Lie algebra. Then $\dim V_1\geq32$, and if $\dim V_1\geq 56$ then the root system is among the following:

Theorems & Definitions (44)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 34 more