OZ-Type Vertex Operator Algebras

• OZ-Type Vertex Operator Algebras are graded structures with V₀ = ℂ1, V₁ = 0, and a nonassociative Griess algebra (V₂) generated by simple Virasoro vectors.
• Their rigorous classification relies on fusion relations, mode-product constraints, and invariant bilinear forms that uniquely determine the entire algebraic structure.
• The automorphism group is isomorphic to the symmetric group Sₙ, linking these VOAs to moonshine phenomena and modular invariance in chiral conformal field theories.

An OZ-Type Vertex Operator Algebra (VOA) is a simple, graded algebraic structure of “moonshine type” distinguished by a trivial weight-one subspace ($V_1 = 0$) and a unique vacuum (%%%%1%%%%), with the weight-two space ($V_2$), called the Griess algebra, as its primary nontrivial component. The class considered in (Feng, 10 Oct 2025) consists of VOAs generated by a finite family of simple Virasoro vectors $$\{\omega^{ij} = \omega^{ji}\mid 1 \leq i < j \leq n\}$$ for $n \geq 3$, enforcing strong algebraic constraints that completely determine the VOA's structure, automorphism group, and unitarity properties.

1. Structural Definition and Generating Virasoro Vectors

This class of VOAs has a grading $V = \bigoplus_{k=0}^\infty V_k$ where $V_0 = \mathbb{C}\mathbf{1}$, $V_1 = 0$, and $V_2$ is a commutative, nonassociative algebra called the Griess algebra. The generators $\omega^{ij}$ are simple Virasoro vectors, each spanning a Virasoro subalgebra isomorphic to $L(c_m,0)$ with central charge

$c_m = 1 - \frac{6}{(m+2)(m+3)}.$

The conformal weights are specified by

$$h^{(m)}_{r,s} = \frac{([r(m+3) - s(m+2)]^2 - 1)}{4(m+2)(m+3)},$$

in particular $h^{(m)}_{m+1,1} = \frac{m(m+1)}{4}$.

These Virasoro vectors form the sole generators, satisfying the fusion relations

$$\omega^{ij}_1 \omega^{jl} = \frac{1}{2} h^{(m)}_{m+1,1}\left(\omega^{ij} + \omega^{jl} - \omega^{il}\right),$$

and high-mode products, such as $\omega^{ij}_p \omega^{kl} = 0$ for $p \geq 0$ when indices are distinct, enforce further algebraic rigidity.

The full conformal vector $\omega$ of $V$ is a fixed linear combination: $$\omega = \frac{8}{(n-2)m(m+1)+8} \sum_{1 \leq i < j \leq n} \omega^{ij}.$$ This uniquely specifies the Virasoro structure for the entire algebra.

2. Spanning and Uniqueness via the Griess Algebra

Every element of $V$ is expressible as a linear combination of iterated modes applied to the vacuum: $$\omega^{i_1j_1}_{n_1}\omega^{i_2j_2}_{n_2}\cdots\omega^{i_sj_s}_{n_s}\mathbf{1}, \quad n_k \leq 0.$$ Theorem 3.1 (Feng, 10 Oct 2025) establishes that these generators span $V$.

The fundamental uniqueness property is that the VOA's entire algebraic structure is determined by the Griess algebra $V_2$. Theorem 3.2 proves that any VOA with a Griess algebra satisfying these prescribed relations—particularly those between the $\omega^{ij}$—must be isomorphic to $V$. This reduces the classification to studying $V_2$, a finite-dimensional algebra, and is a hallmark of OZ-type structure.

3. Automorphism Group: The Symmetric Group $S_n$

The automorphism group is isomorphic to the symmetric group $S_n$ on $n$ letters. The construction exploits Miyamoto involutions $\sigma^{ij}$, acting as: $$\sigma^{ij}|_{V_{[0]}} = \mathrm{id}, \qquad \sigma^{ij}|_{V_{[h^{(m)}_{m+1,1}]}} = -\mathrm{id},$$ where $V_{[h]}$ are isotypical components under $L(c_m,0)$ generated by $\omega^{ij}$. These involutions implement transpositions among the generators ($\omega^{ij}_1 \omega^{jl}$ swaps $j \leftrightarrow l$, etc.), and their group-theoretic commutation and braid-type relations conform precisely to $S_n$. Consequently, all automorphisms permute the generator set, and no larger group is compatible with the defining VOA relations [(Feng, 10 Oct 2025), Thm 3.4].

4. Explicit Algebraic Relations and Mathematical Formulations

The algebraic backbone of this class is the fusion and mode relations among Virasoro vectors:

• For distinct $i, j, l$:

$$\omega^{ij}_1 \omega^{jl} = \frac{1}{2}h^{(m)}_{m+1,1}(\omega^{ij} + \omega^{jl} - \omega^{il}),$$

• Vanishing higher modes:

$$\omega^{ij}_p \omega^{kl} = 0 \quad (p \geq 0, (i,j)\neq(k,l)).$$

• Norm and inner products in $V_2$:

$$(\omega^{ij}|\omega^{ij}) = \frac{1}{2}c_m, \qquad (\omega^{ij}|\omega^{jl}) = \frac{1}{8}c_m h^{(m)}_{m+1,1},$$

with explicit numeric forms (e.g., $m=2$ yields $c_2 = \frac{7}{10}$).

These relations ensure that, once $V_2$ is specified, all higher VOA relations and module structures are rigidly fixed.

5. Unitarity Constraints on Parameters

Unitarity requires the existence of a positive-definite Hermitian form compatible with the VOA structure and the conjugate linear involutions. For the generators,

$$(\omega^{ij}|\omega^{ij}) = \frac{m(m+5)}{2(m+2)(m+3)}, \qquad (\omega^{ij}|\omega^{jl}) = \frac{m^2(m+1)(m+5)}{32(m+2)(m+3)},$$

and the Gram matrix formed by these inner products must be positive-definite.

The determinant analysis yields explicit restrictions:

• For $n=3$, $m \leq 3$ is necessary,
• For $n \geq 4$, only $m=2$ yields a unitary algebra.

Therefore, only specific values of $m$ and $n$ can produce an OZ-type VOA in this family that is unitary.

6. Context and Implications

The methodology implemented leverages spanning the VOA using the low-weight subspace, enforcing mode-product relations, and using the invariant bilinear form and Miyamoto involutions to capture automorphism symmetries (Feng, 10 Oct 2025). This mirrors group-theoretic constructions in moonshine-type VOAs, where the Griess algebra often reflects significant underlying symmetry (Monster, symmetric group, etc). The realization that unitary representatives exist only for tightly constrained parameter values indicates strong rigidity. This class therefore provides concrete models for studies of moonshine phenomenon, modular invariance, and chiral CFT classification programs, and is directly relevant for understanding holomorphic and self-dual chiral algebras.

Table of Key Structural Data

Generator Structure	Defining Relation	Automorphism Group
$\omega^{ij}$, $1\leq i < j \leq n$	$$\omega^{ij}_1 \omega^{jl} = \frac{1}{2} h^{(m)}_{m+1,1} (\omega^{ij} + \omega^{jl} - \omega^{il})$$	$S_n$

The entirety of the VOA's higher weight structure, correlation functions, automorphism group, and unitarity can be deduced from the Gram matrix and fusion relations in $V_2$, underscoring the significance of the Griess algebra in OZ-type vertex operator algebra theory.