Classification and Comparison Problems across QFT Frameworks

QFT frameworks encode different objects: distributions of fields, Euclidean correlation hierarchies, operator-algebraic nets, locally covariant functors, cochain-valued factorization algebras, vertex operator algebras, conformal nets, and bordism functors. A comparison theorem must name its source and target categories, construct a functor, identify what it preserves, and state whether it is faithful, fully faithful, essentially surjective, or an equivalence. Matching benchmark observables is not a replacement for these maps.

Required background. Factorization comparison theorems supply a Lorentzian model. Partial net-to-VOA reconstruction supplies an analytic converse problem. Extended-TQFT classification scope supplies a higher-categorical target, and existence, uniqueness, and equivalence fix the logical vocabulary. Helpful background. DHR reconstruction and the framework comparison in Foundations provide two further directions.

A directional comparison matrix

Every row should answer the same questions.

Comparison	Source object	Target object	Proved direction	Missing upgrade
OS reconstruction	Reflection-positive Euclidean hierarchy with growth and symmetry	Wightman theory up to unitary equivalence	Euclidean $\to$ Lorentzian	Unrestricted converse and equivalence of arbitrary presentations
AQFT–prefactorization	Additive time-slice AQFT or time-orderable prefactorization algebra in the stated target	The corresponding object on the same Lorentzian site	One-categorical equivalence under explicit hypotheses	General cochain-valued infinity-categorical localization
VOA $\to$ net	Simple unitary energy-bounded strongly local VOA	Irreducible diffeomorphism-covariant conformal net	Canonical net and recovery for the constructed family	Every unitary VOA strongly local; every conformal net in the essential image
Net $\to$ VOA	Conformal net with finite-energy generating fields and energy bounds	Strongly local unitary VOA	Conditional reconstruction	Universal reconstruction without point-field hypotheses
Cobordism classification	Fully dualizable object in a named symmetric monoidal higher category	Fully extended framed TQFT	Classification in fixed dimension, structure, depth, and target	Classification of arbitrary metric-dependent QFTs

The matrix forbids undirected language. An arrow can preserve local observables but not positive states; a quasi-isomorphism can preserve cohomology but not a $C^*$-completion; a fully faithful functor can miss target objects.

Three comparison theorems

Osterwalder–Schrader reconstruction starts with a full Euclidean Green-function hierarchy satisfying reflection positivity, Euclidean covariance, symmetry, clustering, regularity, and growth assumptions. It constructs the Hilbert space, vacuum, translations, and Wightman distributions Osterwalder and Schrader 1975, §§4–6, pp. 291–305. The theorem does not say that arbitrary formal Euclidean correlators satisfy those hypotheses.

For Lorentzian AQFT and time-orderable prefactorization algebras, Benini, Carmona, Grant-Stuart, and Schenkel prove an equivalence for additive theories satisfying the relevant time-slice conditions when the target is a bicomplete closed symmetric monoidal one-category 2024, Theorems 3.3–3.4. For cochain-complex targets, their reduction theorem isolates a spacetime-wise infinity-localization question, but the available detection criteria remain inconclusive 2024, Open Problem 5.6, Proposition 5.7, and Remark 5.8. This dated distinction prevents a strict theorem from being advertised as a universal derived equivalence.

For a simple unitary energy-bounded strongly local VOA $V$, the closed smeared vertex operators generate an irreducible conformal net $\mathcal A_V$ Carpi, Kawahigashi, Longo, and Weiner 2018, Theorem 6.8, PDF pp. 52–53. Their Fredenhagen–Jörß construction recovers $V$ from $\mathcal A_V$ 2018, Theorem 9.2, PDF pp. 67–68. A converse for an independent net needs finite-dimensional energy eigenspaces and a PCT-stable, energy-bounded family of quasi-primary generators 2018, Theorem 9.3, PDF pp. 68–71. The two statements are not a universal net–VOA equivalence.

First application: chiral blocks and sewing

Return to chiral blocks, sewing, and modularity. Compare three arrows using identical fields.

1. VOA to net: source $V$, target $\mathcal A_V$, hypotheses of unitarity, energy bounds, and strong locality, with interval algebras preserved.
2. OS reconstruction: source Schwinger hierarchy, target Wightman theory, with positivity and analytic/growth hypotheses.
3. AQFT to prefactorization: source additive time-slice net, target time-orderable prefactorization object, with target category and localization class fixed.

Agreement of vacuum characters, fusion coefficients, or low-point correlators can check a proposed comparison. It does not prove essential surjectivity or reconstruct the analytic domains of smeared fields. In CFT, sewing data may produce consistent formal blocks without proving a unitary Hilbert completion or a conformal net realization.

Extended TQFT is a different classification problem

The cobordism hypothesis identifies fully extended framed TQFTs with fully dualizable objects in the declared target higher category, together with the appropriate homotopy fixed-point data for added tangential structure. It does not take a Wightman theory as input, forget the metric, and return an equivalent TQFT. Schommer-Pries gives a complete two-dimensional extended classification in the bordism bicategory setting and, for the algebra–bimodule target, identifies oriented theories with separable symmetric Frobenius algebras 2011, §3.8, pp. 230–244.

Failure test: three double arrows

Replace the three proved or conditional rows by equivalence symbols because selected observables match. The OS row loses its converse hypotheses; the AQFT row loses additivity, time-slice, target-category, or localization conditions; the VOA row loses strong locality and essential-image control. The strongest surviving claims are the original direction-specific theorems.

An independent check computes the unit and counit. For a claimed equivalence $F\dashv G$, verify $GF\simeq\mathrm{id}$ and $FG\simeq\mathrm{id}$ on every object class in scope. A round trip on one free model is an example, not categorical essential surjectivity.

Exercises

Why does equality of fusion rings not prove equivalence of VOA and conformal-net representation categories?

Solution

The Grothendieck ring forgets associators, braiding, duality maps, $*$-structure, analytic localization, and morphism spaces. Distinct tensor categories can have the same fusion coefficients. A categorical equivalence must construct a functor that preserves the full stated structure and prove full faithfulness and essential surjectivity.

References

• Benini, Marco, Victor Carmona, Alastair Grant-Stuart, and Alexander Schenkel. “On the Equivalence of AQFTs and Prefactorization Algebras.” arXiv:2412.07318 (2024). arXiv.
• 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.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. Project Euclid.
• 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.