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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 $G$-orbifold of a holomorphic (trivial representation category) rational conformal 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 (too tired) it also follows conversely, that if a CFT has $D^\omega(G)$ 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) $D^\omega(G)$

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) $D^\omega(G)$

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 $G$-orbifold of a holomorphic (trivial representation category) rational conformal 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 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 (too tired) it also follows conversely, that if a CFT has $D^\omega(G)$ 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) $D^\omega(G)$

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 $G$-orbifold of a holomorphic (trivial representation category) rational conformal 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 (too tired) it also follows conversely, that if a CFT has $D^\omega(G)$ 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) $D^\omega(G)$

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 $G$-orbifold of a holomorphic (trivial representation category) rational conformal theory is $D^\omega(G)$ for some $[\omega]$, see Corollary 3.6. in http://arxiv.org/abs/0909.2537

But I have now idea if 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 (too tired) it also follows conversely, that if a CFT has $D^\omega(G)$ 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 $D^\omega(G)$

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 $G$-orbifold of a holomorphic (trivial representation category) rational conformal 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 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 (too tired) it also follows conversely, that if a CFT has $D^\omega(G)$ 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) $D^\omega(G)$