Unitary vertex operator algebras and conformal nets

Proves strongly rational unitary vertex operator algebras give conformal nets; claims their representations match.

1 paper September 25, 2026 Explained by AI

Physicists have two rigorous languages for two-dimensional conformal field theories: vertex operator algebras and conformal nets. It was expected that they describe the same theories, but this was proved only family by family. This paper claims the match for every strongly rational unitary vertex operator algebra, including their modules, fusion and braiding. Lean checks the first step, building the net, but not the full equivalence.

First step

—

Carpi, Kawahigashi, Longo and Weiner, who built nets from strongly local VOAs and conjectured strong locality for every simple unitary VOA

Significance

Major

It settles, for the whole strongly rational unitary class, a correspondence that earlier work proved only for specific families.

Trust

Formal proof listed

Field

Mathematical physics

1 paper

The explanation

Take the Ising model at its critical point, a standard example from statistical physics. Its chiral part has three basic sectors, with conformal weights 0, 1/16 and 1/2. Each sector has a quantum dimension, here 1, √2 and 1. Two copies of the 1/16 sector fuse into the other two sectors together. That matches √2 × √2 = 1 + 1 = 2. A net-side invariant called the μ-index is the sum of the squared dimensions, 1 + 2 + 1 = 4. Two different theories should agree on all of this data.

Vertex operator algebras and conformal nets are two mathematical formulations of chiral conformal field theorySource 1. A vertex operator algebra, or VOA, records fields and their algebraic identities. A conformal net assigns algebras of bounded observables to intervals of a circleSource 2. Unitary means the theory has a positive inner product, as quantum physics needs. Strongly rational is a set of finiteness conditions. One of them says every suitable module splits into simple pieces.

Carpi, Kawahigashi, Longo and Weiner made strong locality the condition for building a net from a VOASource 3. Strong locality means fields on disjoint intervals give commuting bounded observables. They conjectured it for every simple unitary VOASource 4. The obstacle is analytic. Fields that commute formally on a common domain need not give commuting algebrasSource 5. The full match was known for several families, such as affine VOAs at positive integer levels and even lattice VOAsSource 6.

The paper claims the correspondence for every simple unitary strongly rational VOASource 7. The first step is proved in Lean. The VOA has polynomial energy bounds and is strongly local, and its smeared fields form an irreducible conformal netSource 8. Energy bounds limit how fast each field grows with energy. Second, every simple module can be given a positive inner product, and the fusion forms are positiveSource 9. Third, the net is completely rational, a finiteness condition on the net sideSource 7. Last, a functor of Carpi, Weiner and Xu matches the two module categories, with their fusion and braidingSource 10. Through Gui's theorems, finite-index local extensions of the nets also match conformal extensions of the VOAsSource 11.

A key step bounds field growth. A finite generating family and two contraction inequalities give polynomial growthSource 12. A sharper estimate then gives the order needed for localitySource 12.

How sure is this? Lean, a computer proof checker, has a formal proof of the first step. It covers polynomial energy bounds, strong locality and the irreducible conformal net. Complete rationality, unitarity of all modules, the tensor equivalence and the extension classification are outside it. Those parts rest on the paper's long argument and much cited work. No specialist review has been published.

What it doesn't show. This does not cover vertex operator algebras that are not strongly rational or not unitary, and Lean does not check the tensor equivalence.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Lean covers energy bounds, strong locality and the net construction; the representation-theory equivalence rests on the paper and cited work, with no published specialist review.

Words used

Vertex operator algebra (VOA): An algebraic structure that encodes the fields of a two-dimensional conformal field theory and how they combine.
Conformal net: An assignment of operator algebras of observables to intervals of a circle, compatible with conformal symmetry.
Unitary: Having a positive inner product compatible with the structure, as quantum physics requires.
Strongly rational: A finiteness condition: CFT type, self-dual, all suitable modules split into simple pieces, and a finite-dimensional quotient V/C₂(V).
Strong locality: Fields smeared on disjoint intervals generate operator algebras that commute.
Energy bounds: Estimates that limit how large a field's modes can make a state, in terms of its energy.
Fusion and braiding: Rules for combining two sectors into a third, and for swapping their order.
Completely rational: A net-side finiteness condition: irreducible, split, strongly additive, with finite μ-index.

The original paper

Strongly rational unitary vertex operator algebras and conformal nets (PDF, opens in a new tab)
September 25, 2026

Lean notes

All quoted lines (12)

Vertex operator algebras and conformal nets are two mathematical formulations of chiral conformal field theory.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
A conformal net assigns von Neumann algebras of bounded observables to intervals of the circle.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
Carpi, Kawahigashi, Longo and Weiner made strong locality the analytic condition for constructing a conformal net from a simple unitary VOA.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
They conjectured strong locality for every simple unitary VOA.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
The locality issue is substantial: commuting unbounded fields on a common domain need not generate commuting von Neumann algebras.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
Complete unitarity, full braided equivalence and complete rationality are already known for affine VOAs at positive integral levels, even positive-definite lattice VOAs, parafermion VOAs at positive integral levels, and type $ADE$ discrete-series $W$ -algebras.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
$V$ has polynomial energy bounds and is CKLW-strongly local.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
Gui's extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.
Strongly rational unitary vertex operator algebras and conformal nets Back to text
Two contraction inequalities first give polynomial growth and then, for Hermitian quasi-primary fields of weight $d\geq2$ , the energy order $d-1$ needed for locality.
Strongly rational unitary vertex operator algebras and conformal nets Back to text

Other sources

openai/math, Lean scope note for family 280 The selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

279Exact quantum factoring over a fixed finite gate setMathematical physics
281QAOA attains the SK optimum in the thermodynamic-first limitMathematical physics
278Failure of Kohn–Sham ensemble representationMathematical physics
282From scale symmetry to local conformal symmetry in four-dimensional QFTMathematical physics
277Threshold repetition for entangled gamesMathematical physics

Browse Mathematical physics Lean-checked results