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Unbounded field operators in categorical extensions of conformal nets

Bin Gui

TL;DR

The work provides a unifying framework to relate unitary VOA tensor categories and conformal net tensor categories by using unbounded left/right operators and weak-to-strong categorical extensions. It proves that Rep^{u}(V) and Rep^{f}(A_V) are equivalent braided C*-tensor categories for broad families of VOAs, including all WZW models, lattice VOAs, parafermion VOAs, and ADE-type W-algebras, and yields a new proof of the complete rationality of the associated conformal nets. The central technical advance is showing that polynomial energy bounds plus the strong intertwining property imply strong braiding, enabling an exact categorical match with Connes fusion and the net’s braiding. This establishes unitary modular tensor category structure and full faithfulness of the natural functor to net representations, thereby unifying VOAs and conformal nets in a broad, practically applicable setting with implications for 3D TQFTs and related algebraic structures. The methods extend CKLW18 by deriving strong braiding from energy bounds and intertwining properties, and they provide concrete procedures (compression, cosets, tensor products) to verify the hypotheses for additional VOA classes. Overall, the results greatly expand the catalog of theories with equivalent VOAs and conformal nets and give new, robust tools to prove complete rationality and modularity in this interplay.

Abstract

We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type $ADE$ discrete series $W$-algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given.

Unbounded field operators in categorical extensions of conformal nets

TL;DR

The work provides a unifying framework to relate unitary VOA tensor categories and conformal net tensor categories by using unbounded left/right operators and weak-to-strong categorical extensions. It proves that Rep^{u}(V) and Rep^{f}(A_V) are equivalent braided C*-tensor categories for broad families of VOAs, including all WZW models, lattice VOAs, parafermion VOAs, and ADE-type W-algebras, and yields a new proof of the complete rationality of the associated conformal nets. The central technical advance is showing that polynomial energy bounds plus the strong intertwining property imply strong braiding, enabling an exact categorical match with Connes fusion and the net’s braiding. This establishes unitary modular tensor category structure and full faithfulness of the natural functor to net representations, thereby unifying VOAs and conformal nets in a broad, practically applicable setting with implications for 3D TQFTs and related algebraic structures. The methods extend CKLW18 by deriving strong braiding from energy bounds and intertwining properties, and they provide concrete procedures (compression, cosets, tensor products) to verify the hypotheses for additional VOA classes. Overall, the results greatly expand the catalog of theories with equivalent VOAs and conformal nets and give new, robust tools to prove complete rationality and modularity in this interplay.

Abstract

We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type discrete series -algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given.

Paper Structure

This paper contains 28 sections, 85 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Equivalence of braided $C^*$-tensor categories
  3. Strong braiding of smeared intertwining operators
  4. Main idea of the proof
  5. Outline of the paper
  6. General theory
  7. Categorical extensions of conformal nets
  8. Closable field operators
  9. The dense subspaces $\mathcal{H}_i^\infty$, $\mathcal{H}_i^\infty(I)$, and $\mathcal{H}_i^w(I)$
  10. Weak categorical extensions
  11. From weak to strong categorical extensions
  12. Weak left and right operators
  13. Weak categorical partial extensions
  14. Application to vertex operator algebras
  15. Polynomial energy bounds
  16. ...and 13 more sections

Key Result

Theorem I

Let $V$ be a (finite) tensor product of the following unitary VOAs: Then the following are true: Moreover if $V$ is a unitary sub-VOA of $U$ which is also a tensor product of the above unitary VOAs, and if $V^c$ (the commutant of $V$ in $U$) is regular, then $V^c$ satisfies (a) (b) (c).

Theorems & Definitions (158)

  • Theorem I
  • Theorem II: Main theorem
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3: Gui21a theorem 3.13
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 148 more