# $\text{Rep}(D(G))$ as representation category of a vertex operator algebra

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?

This answer is related to my answer here: Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles) 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) 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

• The following link for reference [1] (Barron-Dong-Mason) has no paywall: arxiv.org/abs/math/9803118 – S. Carnahan Mar 10 '14 at 9:43
• Okay, I've had a look at the references, and I have not seen a proof that the cocycle is trivial. It is an assumption in Kirillov's paper [30] (arxiv.org/abs/math/0104242), and is simply claimed to be "known" in the paragraph of Davydov's paper that you quoted. The true fact in Barron-Dong-Mason is that the $\sigma$-twisted module category is semisimple with one irreducible object for any permutation $\sigma$ (and they give an explicit construction of the irreducible twisted module). – S. Carnahan Mar 10 '14 at 10:43
• I could neither find aproof about my last statement in the literature. But I would still bet it is true! – Marcel Bischoff Mar 10 '14 at 14:33
• In Section 4.2. Davydov also refers to [30] for a proof. – Marcel Bischoff Mar 11 '14 at 9:33
• I agree that the last statement is likely to be true, but I am fairly certain that it is still open for $n \geq 3$, and that Davydov's claims are a bit too optimistic. – S. Carnahan Mar 11 '14 at 12:11

A lot has happened in the last four years, and we now have lots of positive results.

The current state of knowledge is given in Evans-Gannon, "Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds". In particular, if $G$ is a finite solvable group, then $D(G)$ (and more generally, any twist $D^\omega(G)$) is the representation category of some vertex operator algebra (in particular, the fixed points of a $G$-action on some holomorphic vertex operator algebra).

For non-solvable groups, the result you want would follow from the conjectured regularity of fixed points (i.e., a suitable generalization of C-Miyamoto).

Oddly enough, it turns out that permutation orbifolds can have non-trivial twists. This is discussed in the Evans-Gannon paper, and earlier in Johnson-Freyd's "The Moonshine Anomaly".