Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator algebra (VOA). Is the global gauge symmetry under the Doplicher-Roberts (DR) reconstruction a Hopf algebra? How do I understand the global gauge symmetry in the language of the MFC or the VOA?
For a general net in two spacetime dimensions, the DHR category is generally a braided tensor category. Let me assume that it is a braided fusion category. Is the global gauge symmetry a quasi-triangular quasi-Hopf algebra? Can the global gauge symmetry be understood as arising from a fusion category? Relatedly, given a net of subfactors $N \subset M$, can I interpret $N$ as the observable algebra and $M$ as the field algebra? Naively, I want to say that the global gauge symmetry is the fusion category $C$ associated with the subfactor $N \subset M$, but $\mathrm{Rep}(C)$ is not a tensor category. Or is the global gauge symmetry the tube algebra?
In particular, consider putting two completely rational conformal nets together into a two-dimensional net. Presumably, the DHR category is the Drinfeld center of the MFC. Is the global gauge symmetry the corresponding tube algebra, and the field net arising from $\bigoplus_{i,j} V_i \otimes \bar V_j$?
Note: I am a physicist with some familiarity with vertex operator algebras and tensor categories, but new to algebraic quantum field theory. Any help is greatly appreciated.
