Direction-Specific VOA–Net Extension Theorems and Open Converses

VOA extensions and conformal-net extensions agree in proved analytic and categorical regimes, but the correspondence is direction-specific. A recent comparison theorem starts with a completely unitary, strongly energy-bounded, strongly local VOA satisfying a strong-intertwining condition; it proves that each unitary VOA extension is strongly local and that its CKLW net equals the local Q-system extension obtained from the corresponding Frobenius algebra. It does not follow from matching fusion coefficients or modular matrices, and it is not a universal equivalence for arbitrary VOAs and nets.

Required background. VOA Extensions, Orbifolds, Cosets, and Commutants supplies algebra objects. From VOAs to Conformal Nets supplies the net construction. From Nets to VOAs: Partial Reconstructions and Limits supplies the reverse construction and its current boundary.

Helpful background. Extensions, Orbifolds, Cosets, and Alpha-Induction supplies Q-systems. Conformal-Net Classification: Invariants and Limits separates invariants from classification. Duality Claims, Dictionaries, Regimes, and Evidence gives a general discipline for directional claims. Four-Dimensional Chiral-Algebra Sectors and Lost Information supplies a protected-sector contrast.

The extension comparison theorem

Let $V$ be completely unitary, strongly energy-bounded, and strongly local. Gui’s Condition II further asks for unitary $V$-modules that tensor-generate $\operatorname{Rep}^u(V)$ and for the relevant intertwining operators to be energy-bounded and to satisfy the strong intertwining property with disjointly localized VOA fields. This condition is verified for unitary affine VOAs, even lattice VOAs, ADE discrete-series $W$-algebras, parafermion VOAs, their tensor products, and specified rational cosets; the precise family and stronger Condition I distinctions are stated in Gui 2026, Definition 4.15, Theorem 4.20, and Example 6.19, PDF pp. 41–42 and 70–71.

Let $U\supset V$ be a conformal VOA extension with the same conformal vector. Its $V$-module object determines a haploid commutative $C^*$-Frobenius algebra $P$ in $\operatorname{Rep}^u(V)$. The braided $*$-functor from unitary $V$-modules to finite-index representations of $\mathcal A_V$, including its Wassermann tensorator, pushes $P$ to a local Q-system $\Theta$. There are then two constructions:

$$U\longmapsto\mathcal A_U, \qquad P\longmapsto\Theta\longmapsto\mathcal B_\Theta\supset\mathcal A_V.$$

Under Condition II, every unitary $U$-module is strongly integrable and $U$ is strongly local Gui 2026, Theorem 6.2, PDF pp. 59–60. The main comparison theorem identifies the two net extensions on the same Hilbert space,

$$\mathcal A_U=\mathcal B_\Theta,$$

and gives a commutative comparison of module functors; when $U$ also satisfies Condition II, the comparison includes braided tensorators Gui 2026, Theorem 6.11, PDF pp. 65–67. The proof works by identifying smeared charged intertwining operators with the left and right operators of the categorical extension, then showing that both constructions generate the same local von Neumann algebras.

A lattice simple-current fixture

The first application is a concrete instance of the extension comparison used by Chiral Blocks, Sewing, and Modular Invariance. In a one-dimensional Euclidean lattice space, take

$$L=\sqrt8\,\mathbb Z\subset M=\sqrt2\,\mathbb Z.$$

Both lattices are even and $[M:L]=2$. The nontrivial coset $\lambda+L$, $\lambda=\sqrt2$, determines an order-two simple current $J=V_{\lambda+L}$ of $V_L$. Its lowest conformal weight is $h_J=(\lambda,\lambda)/2=1$, so its twist is trivial. The VOA extension is

$$V_M\cong V_L\oplus J.$$

Even-lattice VOAs satisfy the comparison theorem’s strong hypotheses. The algebra object $P=\mathbf1\oplus J$ is commutative, and its pushforward Q-system gives a local index-two extension $\mathcal B_\Theta$ of $\mathcal A_{V_L}$. Gui’s theorem yields

$$\mathcal A_{V_M}=\mathcal B_\Theta.$$

The vacuum Hilbert space restricts as $\mathcal H_{V_M}=\mathcal H_{V_L}\oplus\mathcal H_J$, so the sector branching agrees on both sides. As an independent check, the Jones index is the squared categorical dimension of $P$: $d(P)=1+d(J)=2$ for an invertible $J$, while the extension inclusion has index $d(P)=2$ in the local-Q-system convention. This matches the lattice index and distinguishes the dimension of the algebra object from its square where alternative subfactor conventions are used.

Nonconverses and missing uniqueness

Equal modular $S,T$ matrices do not determine a braided tensor category, a specific commutative algebra object, its multiplication, or the analytic tensorator used to push it to a Q-system. Even equivalence of representation categories does not by itself identify interval algebras. Therefore a bidirectional arrow drawn only from modular data must be deleted.

The proven direction begins with $V$, the analytic Condition II data, an extension $U$, and a specified braided $*$-functor. It concludes strong locality and equality with the corresponding Q-system net extension. The reverse assertion that every local extension of every conformal net arises from a VOA extension is not contained in this theorem. Nor does it establish full faithfulness or essential surjectivity for all unitary VOAs, all net sectors, nonunitary logarithmic theories, or infinite-index extensions. The 2025 net-to-VOA construction supplies an associated VOA for a broad class of nets, but compatibility with a chosen Q-system and uniqueness of the extension multiplication remain additional comparisons.

Adversarial comparison

Take two examples with the same modular matrices and declare their extension theories equivalent. The test fails unless one constructs the relevant braided functor, transports the precise Frobenius algebra multiplication and unit, proves locality/strong integrability, and identifies the generated algebras. The strongest surviving statement is equality of the recorded modular invariant. Sector branching, extension existence, and net equality remain unproved.

Exercises

1. Verify that the lattice simple current above has order two and trivial twist.

Solution

Twice $\lambda=\sqrt2$ is $2\sqrt2=\sqrt8\in L$, so $2(\lambda+L)=L$. Its conformal weight is $\lambda^2/2=1$, hence $e^{2\pi ih_J}=1$.

2. Name the extra datum absent from fusion coefficients but needed to compare extensions.

Solution

One needs a commutative Frobenius algebra multiplication and unit, together with the braided tensorator that transports them to a Q-system. Fusion coefficients specify only possible summands and multiplicities.

References

• Gui, Bin. “Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets.” arXiv:2505.03235v3 (2026). Open PDF.
• Henriques, André G., and James E. Tener. “Every Conformal Net Has an Associated Unitary VOA.” arXiv:2507.20735v3 (2025). Open PDF.