Vertex Operator Algebra L(21/22,0) ⊕ L(21/22,8)

• The algebra L(21/22,0) ⊕ L(21/22,8) is a uniquely determined simple-current extension of the Virasoro minimal model with central charge 21/22, defined by conformal weights 0 and 8.
• It employs coset techniques and modular tensor category arguments to establish unitarity through a positive-definite Hermitian form and PCT invariance.
• The structure admits exactly 15 irreducible modules with factorized fusion rules derived from admissible Virasoro triples and Ising model fusion, ensuring complete module classification.

The vertex operator algebra (VOA) $L(21/22, 0)\oplus L(21/22, 8)$ is a uniquely determined simple-current extension of the Virasoro minimal model at central charge $c=21/22$. This structure, denoted $\mathcal{M}$, is distinguished by its unitarity, explicit module classification, and factorized fusion algebra. The construction, properties, and computations associated with $\mathcal{M}$ emerge from coset techniques and modular tensor category arguments, particularly in connection with the 3C-algebra and the Ising model fusion rules (Xiangyu et al., 1 Jan 2026).

1. Definition and Basic Structure

The algebra $\mathcal{M}$ arises as the direct sum of two irreducible Virasoro modules: $V = L\left(c_9,0\right)\oplus L\left(c_9,8\right),$ with $c_9 = 1-\frac{6}{(9+2)(9+3)} = 21/22$.

The minimal model $L(21/22,0)$ features irreducible highest-weight modules of the form $L(c_9,h^9_{r,s})$, where

$$h^9_{r,s} = \frac{(r\cdot12 - s\cdot11)^2 - 1}{4 \cdot 11 \cdot 12},$$

with the particular cases $h^9_{1,1}=0$ and $h^9_{1,7}=8$ identifying the conformal weights for the two summands of $\mathcal{M}$.

$\mathcal{M}$ constitutes the unique simple-current extension of $L(21/22,0)$ by the order-two module $L(21/22,8)$. This renders $\mathcal{M}$ self-dual and fully characterized by its Virasoro summands of conformal weights $0$ and $8$.

2. Unitarity

The proof of unitarity for $\mathcal{M}$ utilizes the lattice VOA $V_{\sqrt{2}E_8}$, equipped with a standard positive-definite Hermitian form and a PCT operator: $$\phi\colon \alpha_1(-n_1)\cdots\alpha_k(-n_k)\otimes e^\alpha \mapsto (-1)^k \alpha_1(-n_1)\cdots\alpha_k(-n_k)\otimes e^{-\alpha}.$$ The invariant property

$$\left(Y\left(e^{zL(1)}(-z^{-2})^{L(0)}a,z^{-1}\right)u,\,v\right) = \left(u,\,Y(\phi(a),z)v\right)$$

holds for all $a, u, v$.

For coset constructions, when one considers the commutant of conformal vectors, such as $U_{3C}=\mathrm{Com}_{V_{\sqrt2E_8}}(s)$, every coset VOA inherits this positive-definite Hermitian structure and PCT invariance. In particular,

$$\mathcal{M} = \mathrm{Com}_{U_{3C}}(\omega_{L(1/2,0)})$$

is thereby unitary.

Every irreducible $\mathcal{M}$-module emerges as a direct summand in the decomposition of an irreducible $U_{3C}$-module, and each inherits a positive-definite invariant Hermitian form ([Dong–Lin], Lemma 3.4 in (Xiangyu et al., 1 Jan 2026)).

3. Classification of Irreducible Modules

The irreducible $\mathcal{M}$-modules are indexed as $$\{\mathcal{M}_{k,\ell}\mid k=0,\dots,4;\ \ell=0,1,2\}$$, resulting in $15$ inequivalent modules. This module categorization is derived from the commutant decomposition of the five irreducible $U_{3C}$ modules $U(2k)$, $k=0,\dots,4$, each of which contains three coset types: $$U(2k) \cong L(1/2,0)\otimes\mathcal{M}_{k,0} \oplus L(1/2,1/2)\otimes\mathcal{M}_{k,1} \oplus L(1/2,1/16)\otimes\mathcal{M}_{k,2}.$$

The conformal weights for these modules are:

• For $\ell=0$ (vacuum-type):

$$h(\mathcal{M}_{k,0}) = \begin{cases} 0 & k=0, \ 13/11,~6/11,~1/11,~20/11 & k=1,2,3,4, \end{cases}$$

• For $\ell=1$ ($L(1/2,1/2)$-type):

$$h(\mathcal{M}_{k,1}) = 7/2,~15/22,~35/22,~7/22,~1/22, \text{ for } k=0,\ldots,4,$$

• For $\ell=2$ ($L(1/2,1/16)$-type):

$$h(\mathcal{M}_{k,2}) = 31/16,~21/176,~5/176,~85/176,~133/176,~k=0,\ldots,4.$$

The modular tensor category argument ensures no other irreducible $\mathcal{M}$-modules exist, and the category formed by these $15$ modules is closed under fusion.

4. Fusion Rules and Algebra

The fusion product for irreducible modules of $\mathcal{M}$ is expressed as: $$\mathcal{M}_{i,\ell} \boxtimes \mathcal{M}_{j,\ell'} \cong \bigoplus_{(k,\ell'')} N_{(i,\ell),(j,\ell')}^{(k,\ell'')} \mathcal{M}_{k,\ell''},$$ where the structure constants factor according to: $$N_{(i,\ell),(j,\ell')}^{(k,\ell'')} = N^{U_{3C}}_{i,j}{}^{k} \cdot N^{L(1/2,0)}_{\ell,\ell'}{}^{\ell''}.$$

• $N^{U_{3C}}_{i,j}{}^k$ are the fusion rules of the 3C-algebra, with $U(2i)\boxtimes U(2j)$ summing over $k$ such that the triple $((2i-1,1), (2j-1,1), (2k-1,1))$ is admissible in the minimal model $(A_{10}, E_6)$.
• $N^{L(1/2,0)}_{\ell,\ell'}{}^{\ell''}$ correspond to the fusion rules of the Ising model.

Explicitly:

• If $\ell=0$ or $\ell'=0$, $\mathcal{M}_{i,0}\boxtimes \mathcal{M}_{j,\ell'}$ is the sum over admissible $k$ of $\mathcal{M}_{k,\ell'}$.
• If $\ell=\ell'=1$, $\mathcal{M}_{i,1}\boxtimes\mathcal{M}_{j,1}$ is the sum over admissible $k$ of $\mathcal{M}_{k,0}$.
• If $\ell=1$, $\ell'=2$, $\mathcal{M}_{i,1}\boxtimes\mathcal{M}_{j,2}$ is the sum over admissible $k$ of $\mathcal{M}_{k,2}$.
• If $\ell=\ell'=2$, $\mathcal{M}_{i,2}\boxtimes\mathcal{M}_{j,2}$ is the sum over admissible $k$ in $\mathcal{M}_{k,0} \oplus \mathcal{M}_{k,1}$.

“Admissible” refers to those $k$ for which the Virasoro triple is permitted.

5. Modular Tensor Category and Verlinde Formula

As a rational and $C_2$-cofinite VOA, $\mathcal{M}$ permits fusion rule computations via the Verlinde formula: $$N_{ij}{}^k = \sum_\alpha \frac{S_{i\alpha} S_{j\alpha} \overline{S_{k\alpha}}}{S_{1\alpha}},$$ where $S_{i\alpha}$ is the modular $S$-matrix for $\mathcal{M}$. By construction, $S$ for $\mathcal{M}$ factorizes into the tensor product of the Virasoro $S$ and the 3C-algebra $S$.

The fusion rules determined categorically through coset and simple-current arguments directly yield the factorized fusion structure, circumventing the explicit inversion of the full $S$-matrix.

6. Summary of Key Properties

• $\mathcal{M}=L(21/22,0)\oplus L(21/22,8)$ features central charge $c=21/22$.
• It is a unitary VOA: positive-definite Hermitian form and PCT invariance are inherited from its embedding in the coset construction.
• There exist exactly 15 irreducible $\mathcal{M}$-modules, $\mathcal{M}_{k,\ell}$, with precisely determined conformal weights.
• The fusion algebra is completely prescribed by admissible Virasoro triples $(A_{10},E_6)$ and fusion rules of the $L(1/2,0)$ Ising model.

All points above are substantiated by the construction, argumentation, and computations presented by Xiangyu Jiao and Wen Zheng in "A unitary vertex operator algebra arising from the 3C-algebra" (Xiangyu et al., 1 Jan 2026).