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Constructing equivalences between fusion categories of quantum groups and of vertex operator algebras via quantum gauge groups

Claudia Pinzari

TL;DR

The work addresses constructing a direct braided tensor equivalence between fusion categories from quantum groups at roots of unity and those from affine vertex operator algebras at positive integer levels. It develops a quantum gauge-group framework via discrete weak Hopf algebras with a Drinfeld twist to align braiding, associativity, and unitary structure with the Huang–Lepowsky VOA tensor theory, yielding full (A,B,C,D,G2) equivalences and partial results for E,F. Central to the approach are the Wenzl fundamental representation, Zhu’s algebra, and a systematic transfer of modular data through a weak quasi-Hopf/weak Hopf formalism, enabling transport of unitarity and rigidity between the two settings. The results illuminate a unified perspective on quantum group and VOA dualities, providing concrete mechanisms to transport tensorial, modular, and unitary structures, with applications to conformal nets, Doplicher–Roberts reconstruction, and potential extensions to broader CFT models.

Abstract

We explain the structure of proof of our results on a problem that Sergio Doplicher posed in the 90s in the setting of conformal nets, and on a related problem posed by Yi-Zhi Huang in the setting of vertex operator algebras. The latter asks to find a direct construction of an equivalence due to Finkelberg between the quantum group modular fusion category at certain roots of unity and the fusion categories of affine vertex operator algebras at positive integer levels. The former asks for the construction of a quantum gauge group and field algebra for the category of localized endomorphisms of a conformal net, extending Doplicher-Roberts theory of the compact gauge group and field algebra in high dimensional AQFT. We mostly focus on the affine VOA aspects. We solve Huang's problem for all the Lie types for which it is known that the centralizer algebra of the truncated tensor powers of the given generating object V listed by Hans Wenzl in 1998 for the quantum group fusion categories is generated by the representation of braid group associated to the R-matrix and for Huang-Lepowsky ribbon braided tensor structure. In the classical type A, this property becomes the classical Schur-Weyl duality and was used by Doplicher and Roberts to construct SU(d) as a special case of their unique compact gauge group and field algebra in algebraic QFT. Presently, known results in the literature allow to construct the full equivalence for the Lie types A, B, C, D, G_2 and partial equivalence for braiding and associativity morphisms in the Lie types E and F. We also solve a question posed by Frenkel and Zhu in 1992 on the quantum group structure on the Zhu algebra for all Lie types. Our main tool is the construction of a quantum gauge group and a Drinfeld twist method. The structures we find in both contexts extend Doplicher-Roberts symmetric functors which played a central role in their duality theory.

Constructing equivalences between fusion categories of quantum groups and of vertex operator algebras via quantum gauge groups

TL;DR

The work addresses constructing a direct braided tensor equivalence between fusion categories from quantum groups at roots of unity and those from affine vertex operator algebras at positive integer levels. It develops a quantum gauge-group framework via discrete weak Hopf algebras with a Drinfeld twist to align braiding, associativity, and unitary structure with the Huang–Lepowsky VOA tensor theory, yielding full (A,B,C,D,G2) equivalences and partial results for E,F. Central to the approach are the Wenzl fundamental representation, Zhu’s algebra, and a systematic transfer of modular data through a weak quasi-Hopf/weak Hopf formalism, enabling transport of unitarity and rigidity between the two settings. The results illuminate a unified perspective on quantum group and VOA dualities, providing concrete mechanisms to transport tensorial, modular, and unitary structures, with applications to conformal nets, Doplicher–Roberts reconstruction, and potential extensions to broader CFT models.

Abstract

We explain the structure of proof of our results on a problem that Sergio Doplicher posed in the 90s in the setting of conformal nets, and on a related problem posed by Yi-Zhi Huang in the setting of vertex operator algebras. The latter asks to find a direct construction of an equivalence due to Finkelberg between the quantum group modular fusion category at certain roots of unity and the fusion categories of affine vertex operator algebras at positive integer levels. The former asks for the construction of a quantum gauge group and field algebra for the category of localized endomorphisms of a conformal net, extending Doplicher-Roberts theory of the compact gauge group and field algebra in high dimensional AQFT. We mostly focus on the affine VOA aspects. We solve Huang's problem for all the Lie types for which it is known that the centralizer algebra of the truncated tensor powers of the given generating object V listed by Hans Wenzl in 1998 for the quantum group fusion categories is generated by the representation of braid group associated to the R-matrix and for Huang-Lepowsky ribbon braided tensor structure. In the classical type A, this property becomes the classical Schur-Weyl duality and was used by Doplicher and Roberts to construct SU(d) as a special case of their unique compact gauge group and field algebra in algebraic QFT. Presently, known results in the literature allow to construct the full equivalence for the Lie types A, B, C, D, G_2 and partial equivalence for braiding and associativity morphisms in the Lie types E and F. We also solve a question posed by Frenkel and Zhu in 1992 on the quantum group structure on the Zhu algebra for all Lie types. Our main tool is the construction of a quantum gauge group and a Drinfeld twist method. The structures we find in both contexts extend Doplicher-Roberts symmetric functors which played a central role in their duality theory.

Paper Structure

This paper contains 13 sections, 1 theorem, 3 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Motivation and framework of our work. Constructing braided tensor equivalences via a quantum gauge group given by a discrete weak Hopf algebra and a Drinfeld twist.
  4. The general plan of our proof.
  5. Outline of the construction of a weak quasi-tensor structure on the Zhu algebra $A(V_{{\mathfrak g}_k})$ via Wenzl quantisation isomorphism and Drinfeld twist method.
  6. Main tools to prove (a): How to make ${\rm Rep}(A({{V_{\mathfrak g}}_k}))$ into a braided tensor category with Huang-Lepowsky structure.
  7. A central method in the quantum group and VOA setting: the fundamental generating representation $V$ for all Lie types, following H.Wenzl.
  8. Main tools to prove (b), (c): Analogies and differences between Drinfeld-Kohno Theorem and Theorem 30.4 in CGP
  9. Differences between the original Drinfeld-Kohno theorem and our analogue Theorem 30.4 in the treatment of the associativity morphisms, to prove (d).
  10. Main tools to prove (d): cohomological reduction of a braided tensor structure of a category with a generating object $V$ to braid group duality of $V$
  11. The role of braid group duality for the generating object in determining the full braided tensor structure for the proof of (d).
  12. More remarks from the viewpoint of Doplicher-Roberts reconstruction in AQFT on the relevance of duality for the fundamental representation in the construction of quantum gauge group
  13. Outlook

Key Result

Theorem 2.1

Theorems & Definitions (1)

  • Theorem 2.1