Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
1  
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
1  
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
1  
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 at 4:22
1  
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 at 4:39

1 Answer 1

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) $D^\omega(G)$

share|improve this answer
    
Thanks Marcel for the answer/comment. +1. Let me see. –  Idear Jan 1 at 3:16
    
@ 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 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 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 at 20:10
1  
@ 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 at 1:05

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.