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.

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.