# Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question:

Background:

It is known (in theoretical physics) that the algebraic framework underlying discrete H gauge theories with 2+1D Chern-Simons term is the quasi Hopf algebra $D^\omega(H)$, i.e. the Chern-Simons term introduces a 3-cocycle $\omega \in H^4(BH,\mathbb{Z}) \simeq H^4(H,\mathbb{Z}) \simeq H^3(H,U(1))$ in the cohomology group on the Hopf algebra $D(H)$. People in theoretical physics also call the quasi Hopf algebra $D^\omega(H)$ as another name: twisted quantum doubles, such as A Kitaev's (of Caltech) Annals of Physics 303, 2 (2003), Annals of Physics 321, 2 (2006). The background understanding of these topics (to me) would go to Dijkgraaf-Witten theory original paper.

My question here is inspired by the observation in this arXiv paper published in Nucl.Phys. B. It stated that: "From the point of view of conformal ﬁeld theory it is of interest to mention that the fusion rules of $D^\omega(\mathbb{H}_8)$ for p = 1 coincide with the level 1 SU(2)/($\mathbb{Z}_2 \times \mathbb{Z}_2$)-orbifold (cited a paper by Dijkgraaf, Vafa, Verlinde, Verlinde) after modding out the appropriate $\mathbb{Z}_2$ generated by 1 (see Table 2 here)). Apparently, the algebraic structure of such non-holomorphic orbifolds is still determined by the ‘holomorphic’ Hopf algebra, be it deformed by a non-trivial 3-cocycle. To our knowledge, this has not been noticed before."

A detailed Question:

It seems to me that there may have some duality between: $$\text{quasi Hopf algebra } D^\omega(\mathbb{H}_8) \text{ for p = 1} \leftrightarrow \text{level 1 SU(2)/(\mathbb{Z}_2 \times \mathbb{Z}_2) orbifold}$$ Here $p = 1$ is the 3-cocycles labeled of $H^3(\mathbb{H}_8,U(1))=\mathbb{Z}_8$ for $p$(mod 8) in $\mathbb{Z}_8$. How about other 7 classes other than $p=1$ in $p$(mod 8)?

• Are there other some dualities exist for $$D^\omega(\mathbb{H}_8) \leftrightarrow \text{? orbifold}$$ $$D^\omega(D_8) \leftrightarrow \text{? orbifold}$$ $$D^\omega(\mathbb{Z}_2^3) \leftrightarrow \text{? orbifold}$$ What is the general relation (if any, start with a finite group $H$)? $$D^\omega(H) \leftrightarrow \text{? orbifold}$$

$D_8$ is a dihedral group with 8 group elements. $D^\omega(D_8)$ should have three labels of $p_1$,$p_2$,$p_3$ from $H^3(D_8,U(1))=\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. And $D^\omega(\mathbb{Z}_2^3)$ should have 7 labels of $p_j$ from $H^3(\mathbb{Z}_2^3,U(1))=\mathbb{Z}_2^7$.

ps. Excuse me that my mathematical background is not equivalent to a math PhD (but trained in physics), but this should be a research level question in mathematical physics. Please feel free giving comments/answers. Thank you for all who reply and support!

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The question of finding orbifold constructions of fusion rings is very natural and interesting. On the other hand, I don't see much evidence of a general duality here, since the rings are relatively small. –  S. Carnahan Dec 31 '13 at 22:50
But are there some known examples appear between the twos? e.g. In my post I had provided one example. Examples are fine, it needs not to be very general. Physicists appreciate (many) examples more than a theorem. :) –  Idear Dec 31 '13 at 22:54
See also this post Phy.SE on orbifolds of SU(2)/G' and SO(3)/G' if there are available data in the literature, please let me know. Many thanks! –  Idear Dec 31 '13 at 22:56
What it seems like you mean by duality is that the representation categories of $D^\omega(\mathbb H_8)$ for some $\omega$ and $SU(2)/(\mathbb Z_2 \times \mathbb Z_2)$ are described by the same fusion ring. To clarify then, it seems like what you are asking is this: Given $G,H$ finite groups and $\omega\in H^3(H,U(1))$, is there a G-orbifold theory with fusion ring isomorphic to the fusion ring for $D^\omega (H)$ and if so, what is there relationship? –  Matthew Titsworth Jan 2 '14 at 4:22
The reason that I ask is that below, Marcel uses Muger's result to provide a $G$-orbifold CFT whose representation category is the representation category of $D^\omega(G)$. However, the statement from which you draw your inital observation goes no further than saying that the two are Grothendieck equivalent. –  Matthew Titsworth Jan 2 '14 at 4:39

Regarding the general question, from the point of view of conformal field theory there is a rather trivial way to obtain (some) $D^\omega(G)$. Namely, the representation category of the $G$-orbifold of a holomorphic (trivial representation category) rational conformal field theory is $\mathrm{Rep}(D^\omega(G))$ for some $[\omega]$, see Corollary 3.6. in http://arxiv.org/abs/0909.2537
But I have now idea if as Scott pointed out as a comment all finite groups $G$ (I suppose yes) can be obtained this way; one has to find a holomorphic theory with an action of $G$, for example the Moonshine CFT for the Monster group etc. Then for a given $G$ I also have no idea which $[\omega]$ arise this way.
I think it also follows conversely, that if a CFT has $\mathrm{Rep}(D^\omega(G))$ as representation category, then it is a $G$-orbifold of a holomorphic theory. But then I don't understand the non-holomorphic examples the op mentioned. In the non-holomorphic examples you have to mod something out to become (the dual of) $D^\omega(G)$
@ Marcel: I thought the $D^\omega(H)$ and (if any) its corresponding $G$-orbifold, (such as the example in my post), the $H$ and $G$ are not necessarily the same groups? Are you identifying $H=G$ for some cases? Thanks. –  Idear Jan 1 '14 at 3:21
Any finite group embeds in a sufficiently large symmetric group, and hence in the automorphism group of a sufficiently large tensor product of $E_8$ CFTs. –  S. Carnahan Jan 1 '14 at 10:39
@ S. Carnahan, thanks, can you rephrase automorphism group and $E_8$ statement to the context of our posted question? (I could not fully grasp, are you teaching me something?) –  Idear Jan 1 '14 at 20:10
@ Marcel, do you know any examples of orbifold describing $D^\omega(\mathbb{Z}_2^3)$, $D^\omega(\mathbb{H}_8)$ or $D^\omega(D_8)$? –  Idear Jan 2 '14 at 1:05