DPR orbifold VOA twisted quantum double conjecture

Establish that for any holomorphic orbifold vertex operator algebra V = W^G, where W is a simple vertex operator algebra and G is a finite group of automorphisms of W, there exists a normalized 3-cocycle ω on G such that the weak quasi-Hopf algebra H whose representation category is braided equivalent to Rep(V) can be taken to be the twisted quantum double D^{ω}(G).

Background

The paper reviews how, under suitable finiteness and rationality conditions, the representation category of a vertex operator algebra (VOA) is a modular tensor category. By reconstruction theory, such a category can be realized as the representation category of a (weak) quasi-Hopf algebra H. In the orbifold setting, where V = W^G for a simple VOA W and finite automorphism group G, Dijkgraaf–Pasquier–Roche (DPR) proposed a specific realization using twisted quantum doubles.

The authors cite the DPR conjecture that, in the holomorphic orbifold case, one can choose H to be the twisted quantum double D^{ω}(G) for an appropriate 3-cocycle ω on G. Their work studies gauge equivalence and categorical Morita equivalence involving twisted quantum doubles, which informs when such identifications might hold, but does not resolve the general conjecture.