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

This paper establishes connections between a problem posed by Doplicher and Roberts in the 1990s regarding construction of quantum gauge groups associated to the category of localized endomorphisms of conformal nets, and a problem by Huang concerning a direct proof of the Finkelberg equivalence theorem. We also unify the treatment of high dimensional algebraic QFT with the low dimensional case, resuming the study of Mack-Schomerus quantum symmetries, for which we previously proved a simplifying Hopf algebra property and studied the unitary structure of our global quantum gauge groups A_W({\mathfrak g}, q) of the WZW model. This program leads us to a solution of Huang's problem in the setting of vertex operator algebras. Building upon our previous results, our approach utilizes a quantum gauge group framework to implement a unitary weak quasi-Hopf algebra structure on the Zhu algebra via a Drinfeld twist and Wenzl's de-quantization continuous curve derived from the unitary coboundary weak Hopf algebra A_W({\mathfrak g}, q) previously associated to the quantum group fusion category. Our construction is of an analytic nature, and was originally inspired by Drinfeld-Kohno theorem for Drinfeld category. Using this framework, in our previous work we established an independent rigid braided tensor structure on the module category of the affine vertex operator algebra $V_{{\mathfrak g}_k}$ at positive integer level, providing a braided tensor equivalence with the modular fusion category of the corresponding quantum group at root of unity for all Lie types. In this paper, for the Lie types $A$, $B$, $C$, $D$, and $G_2$ we completely identify our structure with the Huang-Lepowsky braided tensor structure.