A quick Question:

- Is there some duality known between the
quasi Hopf algebra$D^\omega(H)$ of a finite group $H$ to anorbifold model(such as SU(2)/$G$ or SO(3)/$G$ orbifold of some group $G$)? What is this duality relation precisely?

**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!

(many) examplesmore thana theorem. :) – Idear Dec 31 '13 at 22:54