Constructing equivalences between quantum group fusion categories and Huang-Lepowsky modular categories via quantum gauge groups

Abstract

This paper provides a unified framework resolving two long-standing problems: the intrinsic construction of global quantum gauge groups for braided tensor $C^*$-categories (the Doplicher-Roberts problem) and the direct proof of the Finkelberg equivalence theorem at positive integer levels (the Huang problem). In our previous work, we solved both problems for the WZW model across all Lie types by constructing a unitary modular tensor category structure on the module category of an affine vertex operator algebra at positive integer level, together with a quantum gauge group for our analytic structure. Specifically, we utilized the global quantum gauge group A_{W}(\mathfrak{g},q) to equip the Zhu algebra of the affine vertex operator algebra V_{\mathfrak{g}_{k}} with a unitary coboundary weak quasi-Hopf algebra structure with a 3-coboundary associator. This relies on an isometric analytic Drinfeld twist and Wenzl's continuous de-quantization curve. In the present paper, we address the Huang problem specifically for the Huang-Lepowsky tensor structure. We provide a complete identification of our modular tensor category structure with the Huang-Lepowsky structure for all the classical Lie types and $G_2$ via generalized quantum Schur-Weyl duality. This unified approach establishes rigidity directly from the quantum group fusion category, completely bypassing reliance on the Verlinde formula, the monodromy of the Knizhnik-Zamolodchikov equations, negative-level shifting typically required in the VOA setting, and the Jones index used in the conformal net setting. Consequently, our framework restores the natural categorical hierarchy where local rigidity precedes global modular properties entirely within the vertex operator algebra setting.