The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?



$\text{Rep}(D(G))$ as representation category of a vertex operator algebra
This answer is related to my answer here: [Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)](https://mathoverflow.net/questions/153264/duality-between-orbifold-and-quasi-hopf-algebra-twisted-quantum-doubles/153270#153270) and the comment by Scott.


Every finite group $G$ can be embedded in some symmetric group $S\_n$ and then you can take for example 
the $n$-fold product of the $E\_8$ lattice VOA, for which the representation category is trivial. Then I guess the $G$ orbifold of this VOA should have
$\mathrm{Rep}(D^\omega(G))$ as a representation category. This would be true in the setting of conformal net, see my answer here:
[Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)](https://mathoverflow.net/questions/153264/duality-between-orbifold-and-quasi-hopf-algebra-twisted-quantum-doubles/153270#153270)
but I am not sure how much a result like: "an extension of a rational VOA is given by a special symmetric Frobenius algebra $A$ in the category of Representations and the Representation category of the extension is given by the the subcategory of dyslexic (local) module category of $A$." is established in VOA.


In this case the results mentioned in <http://arxiv.org/abs/0909.2537v1> also show that every VOA with $\mathrm{Rep}(D^\omega(G))$ as representation category is a $G$ orbifold of a holomorphic VOA.


**Update** I could still not find an exact statement in VOA, but in 
Alexei Davydov wrote in <http://arxiv.org/abs/1312.7466> (p.2.)



> 
> Our examples come from permutation orbifolds of holomorphic
>  conformal field theories (CFTs 
>  whose state space is an irreducible module over the chiral algebras)
>  . It is argued in [30] (see also [35]) that the modular category of the
>  $G$-orbifold of a holomorphic conformal field theory is
>  the so called
>  Drinfeld
>  (or
>  monoidal
>  )
>  centre
>  $\mathcal Z(G,\alpha)$, where
>  $\alpha$
>  is a 3-cocycle of the group
>  $G$. It
>  is also known that the cocycle $\alpha$
>  is trivial for
>  permutation orbifolds
>  (orbifolds where the group
>  $G$
>  is a subgroup of the symmetric group permuting copies in a tensor power of a holomorphic
>  theory). he assumption crucial for the arguments of [30] is the e
>  xistence of twisted sectors. This
>  assumption is known to be true for permutation orbifolds [1].
> 
> 
> 


This statement (if true, I did not yet check [1] (<http://link.springer.com/article/10.1007/s002200200633>)) gives exactly the answer the ops question, like I guessed above, namely:



> 
> Let $G\subset S^n$ and and let $(V^{\otimes n})^G$ the $G$-permutation orbifold of $V^n$ where 
>  $V$ is a holomorphic VOA, then $\mathrm{Rep}((V^{\otimes n})^G) \cong \mathrm{Rep}(D(G))$.
> 
> 
> 


**Update:** The statement "an extension of a rational VOA is given by a commutative algebra $A$ (Thm 3.2) in the category of Representations and the Representation category of the extension is given by the the subcategory of dyslexic (local) module category of $A$. (Thm 3.4)" appeared now in <http://arxiv.org/abs/1406.3420> 


