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

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 algebraic QFT. In this summary we mostly focus on the affine VOA. We solve Huang’s problem for all the Lie types for which it is known that the centralizer algebras of the non-negligible tensor powers of a given generating object V for the quantum group fusion category (a list of V for which fusion tensor powers are well defined has been given by Hans Wenzl for all the Lie types), is generated by the representation of the braid group associated to the R-matrix, and for Huang-Lepowsky ribbon braided tensor structure for the vertex operator algebras. In the classical type A, this property becomes the classical Schur-Weyl duality for the vector representation, 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 on this generating property allow to construct our equivalence for the Lie types A, B, C, D, G2. Our approach may be regarded in the setting of Connes noncommutative geometry. We also solve a problem by Frenkel and Zhu on the construction of a (unitary) weak quasi-Hopf algebra structure on the Zhu algebra, associated to the minimum energy functor for all the Lie types, equivalent to that from quantum groups. Our main tool is the construction of a unitary coboundary weak Hopf C ∗ -algebra in a sense extending Drinfeld coboundary quasi-Hopf algebras on a formal variable, associated to Wenzl functor for the quantum group fusion category, and of a Drinfeld twist. We apply a de-quantization continuous curve of Wenzl and a Drinfeld twist method to reach the Zhu algebra. The two Drinfeld coboundary symmetries we find in both contexts extend Doplicher-Roberts symmetric functors for rigid symmetric tensor categories with simple unit which play a central role in their duality theory.