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The classification of vertex operator algebras of OZ-type generated by Ising vectors of $σ$-type

Cuipo Jiang, Ching Hung Lam, Hiroshi Yamauchi

TL;DR

This work completes the classification of OZ-type VOAs generated by Ising vectors of σ-type without assuming unitarity. By encoding the structure with finite 3-transposition groups and Matsuo algebras B(G), the authors show the Griess algebra of such VOAs is the non-degenerate quotient of B(G), yielding simple, rational, C2-cofinite, and unitary VOAs whose isomorphism classes correspond to a finite list of groups G_V. The main construction reduces to a hierarchy of cases, including type D (G ≅ F_n: S_n) where V ≅ V_{\sqrt{2}A_{n-1}}^+, and several exceptional cases where V is identified with abelian coset algebras K(R,2) for suitable root systems R. The significance lies in tying VOA classification to finite 3-transposition groups and providing explicit realizations via lattice VOAs, commutants, and abelian cosets, with a compact real form ensuring unitarity. Overall, the paper achieves a complete map from σ-type Ising-vector-generated OZ-VOAs to concrete, well-understood VOA structures and their unitary realizations.

Abstract

We classify vertex operator algebras (VOAs) of OZ-type generated by Ising vectors of $σ$-type. As a consequence of the classification, we also prove that such VOAs are simple, rational, $C_2$-cofinite and unitary, that is, they have compact real forms generated by Ising vectors of $σ$-type over the real numbers.

The classification of vertex operator algebras of OZ-type generated by Ising vectors of $σ$-type

TL;DR

This work completes the classification of OZ-type VOAs generated by Ising vectors of σ-type without assuming unitarity. By encoding the structure with finite 3-transposition groups and Matsuo algebras B(G), the authors show the Griess algebra of such VOAs is the non-degenerate quotient of B(G), yielding simple, rational, C2-cofinite, and unitary VOAs whose isomorphism classes correspond to a finite list of groups G_V. The main construction reduces to a hierarchy of cases, including type D (G ≅ F_n: S_n) where V ≅ V_{\sqrt{2}A_{n-1}}^+, and several exceptional cases where V is identified with abelian coset algebras K(R,2) for suitable root systems R. The significance lies in tying VOA classification to finite 3-transposition groups and providing explicit realizations via lattice VOAs, commutants, and abelian cosets, with a compact real form ensuring unitarity. Overall, the paper achieves a complete map from σ-type Ising-vector-generated OZ-VOAs to concrete, well-understood VOA structures and their unitary realizations.

Abstract

We classify vertex operator algebras (VOAs) of OZ-type generated by Ising vectors of -type. As a consequence of the classification, we also prove that such VOAs are simple, rational, -cofinite and unitary, that is, they have compact real forms generated by Ising vectors of -type over the real numbers.
Paper Structure (25 sections, 34 theorems, 86 equations)

This paper contains 25 sections, 34 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. VOAs generated by Ising vectors of $\sigma$-type
  4. Examples of VOAs satisfying Condition \ref{['cond:1']}
  5. Scaled root lattices.
  6. Commutants and Abelian Coset algebras.
  7. $V_{\sqrt{2}E_8}^+$ and $M_{E_8}$.
  8. Simplicity for type $D$
  9. Irreducible $V^{(n)}$-modules of $\sigma$-type
  10. Lattice VOA $V_{\mathbb{Z}\gamma}$ and its subVOA $V_{\mathbb{Z}\gamma}^+$
  11. $V^{(n)}$ as $V^{(n-1)}\otimes U$-modules
  12. Simplicity of $V^{(n)}$
  13. Simplicity of $V$ for the other cases
  14. Case A:
  15. Case 1: $G=\mathrm{O}_6^-(2)$.
  16. ...and 10 more sections

Key Result

Theorem 1

Let $V$ be a vertex operator algebra of OZ-type generated by Ising vectors of $\sigma$-type. Let $E_V$ be the set of Ising vectors of $V$ of $\sigma$-type and $G_V=\langle \sigma_e\mid e\in E_V\rangle$. Then $G_V$ is finite and the list of the indecomposable pairs $(V,G_V)$ of a VOA $V$ and its grou The $\mathbb{R}$-subalgebra of $V$ generated by $E_V$ is a compact real form of $V$; thus $V$ is un

Theorems & Definitions (46)

  • Theorem 1
  • Lemma 2.2
  • Proposition 2.3: Mi96
  • Definition 2.4: CH
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Theorem 2.10: CH
  • ...and 36 more