Conformal-Net, VOA, TQFT, and Categorical Classification Frontiers

Classification can mean listing isomorphism classes of conformal nets, classifying extensions of a fixed VOA, identifying fully dualizable objects in a chosen higher category, or classifying defect fusion data. These are not interchangeable. A complete algebraic list becomes a QFT classification only after existence, realization, uniqueness, positivity or unitarity, and the relevant comparison functors are proved. This page fixes an evidence cutoff of 2026-08-10 and records several sharply bounded successes beside the remaining realization and converse problems.

Required background. Conformal-net classification invariants and limits supply the operator-algebraic regime. Direction-specific VOA–net extensions supply the current bridge. Extended-TQFT classification scope fixes bordism hypotheses, and categorical classification limits fix defect data. Helpful background. Nonrational modular spectral densities and noninvertible anomalies and RG limits display regimes where finite semisimple classification is insufficient.

Four different classification targets

Target at the cutoff	Proved result	Hypotheses that define its scope	What remains outside
Chiral conformal nets with $c<1$	Complete classification of irreducible diffeomorphism-covariant local nets	Positive-energy conformal nets on $S^1$ with central charge below one	$c\ge1$, nonrational and nonunitary regimes, arbitrary full CFTs
Selected unitary VOA extensions and nets	Strong locality and compatibility of VOA and Q-system extensions	Completely unitary/strongly rational base families satisfying the stated analytic Condition II	Universal strong locality, all nets in the image, nonunitary/logarithmic categories
Fully extended oriented 2D TQFTs in $\mathrm{Alg}_2$	Classification by separable symmetric Frobenius algebras	Two dimensions, full extension, oriented bordisms, algebra–bimodule target	Metric-dependent QFT, arbitrary targets, reflection positivity unless separately imposed
Framed theories from tensor categories	Dualizable finite or fusion tensor categories produce local framed theories at stated depth	Finite tensor category or fusion category with the theorem’s separability/global-dimension assumptions	Classification of all physical phases, oriented/unitary refinements, nonsemisimple realization and completeness

The words “complete classification” apply only to the exact row, not to the union of all neighboring frameworks.

The c < 1 conformal-net theorem

Kawahigashi and Longo classify irreducible diffeomorphism-covariant local conformal nets with central charge $c<1$. The nets correspond to pairs of $A$–$D_{2n}$–$E_{6,8}$ Dynkin diagrams whose Coxeter numbers differ by one, after identifying the permitted local extensions of the Virasoro minimal nets 2004, Theorems 4.1–5.1, pp. 509–520. The proof uses complete rationality, modular invariants, $\alpha$-induction, and subfactor extension theory.

This is a genuine classification of nets in that analytic category. It is not merely a list of modular-invariant matrices. Realization is part of the theorem because the allowed extensions are constructed and identified. The boundary $c<1$ is not decorative: at and above one, continuous families and nonrational phenomena appear, and the finite ADE argument does not give a complete list.

VOA–net realization remains directional

For a simple unitary energy-bounded strongly local VOA $V$, Carpi, Kawahigashi, Longo, and Weiner construct the conformal net $\mathcal A_V$ and recover $V$ from that net 2018, Theorems 6.8 and 9.2, PDF pp. 52–53 and 67–68. This gives an exact round trip on the strongly local image. It does not prove strong locality for every unitary VOA or place every abstract conformal net in that image.

Gui’s current extension theorem advances the bridge for a broad but enumerated set of completely unitary strongly rational bases: unitary affine VOAs, even lattice VOAs, ADE discrete-series $W$-algebras, parafermions, tensor products, and specified cosets under Condition II. Every unitary extension is strongly local, its associated net agrees with the Q-system extension, and the relevant module functors are compared Gui 2026, Theorems 6.2 and 6.11, PDF pp. 59–67. The analytic Condition II and family list are part of the result; “rational VOA” alone is not a replacement.

TQFT classification and the target category

Schommer-Pries presents the oriented and unoriented two-dimensional bordism bicategories and classifies fully extended theories with arbitrary target bicategory. In the symmetric monoidal bicategory of algebras, bimodules, and intertwiners, an oriented theory corresponds to a separable symmetric Frobenius algebra 2011, §3.8, pp. 230–244. Changing the target changes the fully dualizable objects and therefore changes the classification.

In three framed dimensions, Douglas, Schommer-Pries, and Snyder prove that fusion categories of nonzero global dimension are 3-dualizable and hence determine framed local field theories; finite tensor categories are 2-dualizable and determine the corresponding lower-depth theories 2020, Theorems 1–2, pp. 7–12. These theorems construct theories from algebraic objects. They do not assert that every metric-dependent QFT, every nonsemisimple defect category, or every unitary phase is uniquely recovered from a decategorified fusion ring.

First application: one cutoff, four rows

At chiral blocks, sewing, and modularity, compare the following at the same 2026-08-10 cutoff:

1. $c<1$ conformal nets: classified in the theorem’s operator-algebraic category.
2. Rational strongly local VOAs: many important families and extensions map to nets, with current hypotheses explicit; no universal equivalence follows.
3. Fully extended two-dimensional TQFTs: classified for a fixed bordism domain and higher-categorical target.
4. Nonsemisimple defect categories: substantial constructions and invariants exist, but realization, positivity, and completeness depend on the category and physical framework.

A modular tensor category can encode chiral fusion and braiding without selecting a unique full local CFT. Modular data $(S,T)$ are even coarser: inequivalent categories can share such data. Sewing, positivity, and full-center or extension choices remain necessary.

Failure test: classifying data as classifying theories

Count a finite list of fusion rings or Frobenius algebras and announce a classification of all QFTs realizing them. The map from theories to algebraic data may fail to be injective, because distinct theories share the invariant, or fail to be surjective, because some data are not realizable with positivity and locality. The strongest surviving result is classification of the declared algebraic objects.

An independent check has two parts. For each proposed invariant, construct at least one theory realizing it. Then prove that any two theories with the same invariant are equivalent in the stated category. Without both existence and uniqueness, the word “classification” has been enlarged.

Exercises

Why does the cobordism hypothesis not classify four-dimensional Yang–Mills theory?

Solution

The cobordism hypothesis classifies fully extended topological theories for a fixed dimension, tangential structure, extension depth, and symmetric monoidal higher target. Ordinary Yang–Mills depends on a metric and local propagating degrees of freedom and has not been supplied as a fully dualizable object in such a topological target. Forgetting those features changes the theory rather than classifying it.

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.
• Douglas, Christopher L., Christopher Schommer-Pries, and Noah Snyder. Dualizable Tensor Categories. Memoirs of the American Mathematical Society 268, no. 1308 (2020). AMS; Open PDF.
• Gui, Bin. “Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets.” arXiv:2505.03235, revised 2026. arXiv.
• Kawahigashi, Yasuyuki, and Roberto Longo. “Classification of Local Conformal Nets. Case $c<1$.” Annals of Mathematics 160 (2004): 493–522. DOI; Open PDF.
• Schommer-Pries, Christopher J. The Classification of Two-Dimensional Extended Topological Field Theories. PhD thesis, University of California, Berkeley, 2011; revised arXiv version 2014. arXiv.