String operators in the string-net model In the string-net model http://arxiv.org/abs/cond-mat/0404617, quasiparticles are created by the string operators (defined in eq.(19)). An easier pictorial way to define string operators $W_{\alpha}(P)$ is to say that the effect of acting $W_{\alpha}(P)$ on some string-net configuration is to add a type-$\alpha$ string along the path $P$ (in the fattened lattice). One can then use the graphical rules to resolve the added strings into the original (honeycomb) lattice and obtain the resulting string-net configuration living on the original lattice. 
By solving eq.(22), one can find all the string operators. However, only the "irreducible" solutions to eq.(22) give us string operators that create quasiparticle pairs in the usual sense. A generic (reducible) solution to eq.(22) gives us a string operator that creates superpositions of different strings-which correspond to superpositions of different quasiparticles. This point is noted at the end of page 9.
So to analyze a topological phase, and study its quasiparticle excitations, one only needs to find the irreducible solutions $(\Omega_{\alpha},\bar{\Omega}_{\alpha})$ to eq.(22). The number $M$ of such solutions is always finite.
My questions is, in general, how can we find all the irreducible solutions to eq.(22), to obtain the "irreducible" string operators e.g. in eq.(41), (44), (51). I'm particularly interested in the lattice gauge theory case. There is a remark about this case at the end of page 9 and the beginning of page 10: There is one solution for every irreducible representation of the
quantum double $D(G)$ of the gauge group $G$. I'd like to know explicitly, how the solution is constructed if one is given a (finite, perhaps nonabelian) group $G$, and all the irreps of $D(G)$. Lastly, in the Kitaev quantum double framework, quasiparticles are created by the ribbon operators, and there is a known mapping of quantum double models into string-net models, described in http://arxiv.org/abs/0907.2670. Presumably there should be a mapping of ribbon operators into string operators. I'd like to know what that mapping is.
 A: The most general statement one can make about the string operators is that, for a Levin-Wen model constructed from some fusion category $\mathcal{C}$, the irreducible string operators correspond to the simple objects in the braided fusion category $Z(\mathcal{C})$, where $Z$ denotes the Drinfeld center. If one lets $\mathcal{C}$ be the category of representations of a finite group $G$ (which corresponds to the string-net model Levin and Wen give to describe discrete gauge theory), then presumably one finds that $Z(\mathcal{C})$ is the category of representations of the quantum double $D(G)$. However, I am having trouble proving this; hopefully someone else can clarify.
(By the way, the Kitaev quantum double model is itself a string net model, with string types given by group elements $g \in G$, fusion rules given by group multiplication, and trivial $F$ symbols. Presumably the Kitaev ribbon operators are then identical with the Levin and Wen string operators.)
A: The general picture is the following: All Levin-Wen string nets models are conjectured to be derivable from representation categories of weak-Hopf algebras. The most well-known case is the quantum double of finite groups, the Kitaev model. Here the mapping between the ground states are worked out (using Fourier transformation on the group) in the paper quoted in the question and in http://arxiv.org/abs/0907.3724. This latter paper came out 6 days later, but it is an independent work and deserves credit. The next step is when one considers finite dimensional Hopf algebras:
http://arxiv.org/abs/1007.5283. The most general step (weak-Hopf algebras) is done by a PhD student of Zhenghan Wang, it is a tough material to read, I can send the file if someone is interested. (I'll update the thread with his name too later, apologies for not remembering now).
The excitation should coincide automatically (I cannot quote a theorem, but Zhenghan Wang told me), so people didn't bother to work out the details and match the ribbon operators (it is in preparation though:-). The point is that in the formulation of the Levin-Wen string nets all quantities are indexed by labels (of (equivalence classes of) simple objects)) of the unitary tensor category (UTC), but once excitation are described, the distinct particle types correspond to those of the Drinfeld center as Dominic wrote, but the labels of the Omegas in the string operator ansatz are still from the original UTC, so it is a bit hard to follow in detail what's going on. But low rank examples are given, so there must be a general algorithm to work these out. And that algorithm should be based again on the Fourier transformation on the finite group the corresponding Kitaev model is based on.  
