From Nets to VOAs: Partial Reconstructions and Limits

There are now two rigorous net-to-VOA mechanisms with different analytic inputs. The 2018 Fredenhagen–Jörß route reconstructs a strongly local VOA when a conformal net has controlled finite-energy pointlike generators satisfying polynomial energy bounds. A 2025 preprint constructs an AQFT-local unitary VOA from every chiral conformal net in its stated class, whose vacuum rotation generator has finite-dimensional eigenspaces, and proves a round trip back to the original net. This does not prove that every abstract unitary VOA is AQFT-local, nor does it identify all module, sector, or extension categories without further hypotheses.

Required background. Conformal Nets and Covariance Axioms supplies the interval-algebra input. Vertex Operator Algebras: Axioms, Grading, and Locality supplies the target. From VOAs to Conformal Nets supplies the round-trip comparison.

Helpful background. Split Property and Complete Rationality discusses stronger net finiteness properties that are not assumed merely from the existence of a reconstructed VOA.

Fredenhagen–Jörß fields and conditional recovery

Let $\mathcal A$ be an irreducible conformal net on $(\mathcal H,\Omega)$ with conformal Hamiltonian $L_0$, and let

$$\mathcal H^{\mathrm{fin}}=\bigoplus_{n\geq0}\ker(L_0-n).$$

Tomita–Takesaki modular data for $(\mathcal A(I),\Omega)$ associate to a quasi-primary vector $a$ a closed, interval-localized Fredenhagen–Jörß field $Y_I(a,f)$. Compatibility as $I$ varies provides a pointlike field candidate. For a net already obtained from a simple strongly local unitary VOA, these fields coincide with the original smeared vertex operators, so the VOA is recovered on $\mathcal H^{\mathrm{fin}}$ Carpi et al. 2018, Theorem 9.2, PDF pp. 67–68.

The converse theorem in that work is conditional. Assume a family $F$ of quasi-primary vectors generates every interval algebra through its Fredenhagen–Jörß fields, is closed under PCT, satisfies polynomial energy bounds, and every $L_0$ eigenspace is finite-dimensional. Then $\mathcal H^{\mathrm{fin}}$ carries a simple strongly local unitary VOA structure $V$ and $\mathcal A_V=\mathcal A$ Carpi et al. 2018, Theorem 9.3, PDF pp. 68–71. The proof glues fields from an interval cover with a partition of unity, uses energy bounds for a common invariant core, invokes locality to obtain mutually local formal fields, and uses finite multiplicities for the VOA grading restriction.

The annulus construction and its exact reach

Henriques and Tener replace polynomial energy bounds by a geometric construction from the semigroup of annuli. Worm-shaped localized insertions are shrunk to point insertions; their holomorphic sewing identities define vertex products on finite-energy vectors. In version 3 of their preprint, a “conformal net” is understood with finite-dimensional vacuum $L_0$ eigenspaces. Under that convention they prove:

$$\mathcal A\longmapsto V_{\mathcal A}, \qquad V_{\mathcal A}\text{ unitary and AQFT-local}, \qquad \mathcal A_{V_{\mathcal A}}=\mathcal A.$$

See Henriques and Tener 2025, Theorems A–B and Theorem 8.9, PDF pp. 2–3 and 65–66. AQFT-locality means that closed smeared fields with disjoint supports commute strongly; unlike the 2018 definition of strong locality, it does not additionally require polynomial energy bounds. They also prove that applying their net-to-VOA construction to an AQFT-local unitary VOA’s net canonically returns the VOA.

This preprint resolves the net-to-VOA direction for its class more broadly than the earlier generator theorem. The remaining nonconverse is precise: it is still conjectural that every unitary VOA is AQFT-local. Moreover, a net representation yields a module by the preprint’s method only when its rotation generator has discrete spectrum with finite-dimensional eigenspaces. Equality of all unitary-VOA and conformal-net representation theories, and compatibility of arbitrary extensions, are not conclusions of Theorem 8.9.

Virasoro net fixture

The first application is the stress-tensor theory of The Virasoro Algebra and the Stress Tensor. Let $\mathcal A_{\mathrm{Vir},c}$ be the vacuum Virasoro net at an allowed unitary central charge. Its finite-energy space is generated by

$$L_{-n_1}\cdots L_{-n_r}\Omega, \qquad n_i\geq2,$$

modulo the vacuum null relations. The quasi-primary generator $\nu=L_{-2}\Omega$ has Fredenhagen–Jörß field equal to the smeared stress tensor. The Virasoro energy estimate supplies polynomial bounds, and those fields generate the net, so the 2018 conditional theorem reconstructs $L(c,0)$. Reapplying the VOA-to-net construction returns $\mathcal A_{\mathrm{Vir},c}$.

Two independent checks fix the data. Differentiating rotation covariance gives $o(\nu)=L_0$, so the reconstructed grading is the original energy grading. The commutator of reconstructed modes contains $\frac c{12}(m^3-m)\delta_{m+n,0}$, recovering the same central charge. The generated subnet is therefore not merely isomorphic at the level of characters: its interval algebras equal the original net by the recovery theorem. Representation data beyond the vacuum sector still require a separate module–sector comparison.

Adversarial failure test

Take an isotonic family of interval inclusions with no vacuum cyclicity, no positive-energy conformal action, or no finite-dimensional energy spaces, and write formal “modes” from the inclusions alone. This is not an input to either theorem. There are no controlled point insertions, domains, OPE convergence, or grading restriction. Even for a genuine net, an ad hoc choice of generators does not prove that its formal fields generate the full net. What survives is the bounded interval-algebra structure actually supplied; a VOA conclusion needs the hypotheses of the Fredenhagen–Jörß theorem or the annulus construction.

Exercises

1. Explain why finite-dimensional $L_0$ eigenspaces are needed for an ordinary VOA grading.

Solution

An ordinary VOA requires each homogeneous subspace $V_n$ to be finite-dimensional. Taking $V_n=\ker(L_0-n)$ therefore violates the definition if an eigenspace is infinite-dimensional, even when the algebraic direct sum is dense.

2. Which implication remains open after the 2025 preprint?

Solution

Every net in the stated class yields an AQFT-local unitary VOA and returns under the net construction. What remains open is that every abstract unitary VOA is AQFT-local; without that property its conformal net is not defined by this route.

References

• Carpi, Sebastiano, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner. “From Vertex Operator Algebras to Conformal Nets and Back.” Communications in Mathematical Physics 364 (2018), 101–145. DOI. Open PDF.
• Henriques, André G., and James E. Tener. “Every Conformal Net Has an Associated Unitary VOA.” arXiv:2507.20735v3 (2025). Open PDF.