# 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.

• I don't know if this helps, but the solutions for the string operators are basically the R symbols for the corresponding excitations; for discrete gauge theory the theory of the excitations is the representatiom theory of the quantum double and I think the R symbols can be calculated straightforwardly from that. Jan 13 '16 at 18:18
• E.g. see arxiv.org/abs/hep-th/9511201 for an explanation of how the excitations and their braiding and fusion correspond to the representation of the quantum double, which should allow one to calculate the R symbols (though it is not done explicitly in that paper.) Jan 13 '16 at 18:24
• Thanks! So if I understand correctly, for the discrete group case, there should be a 1-1 correspondence between representations of the quantum double of $G$, and the types of irreducible string operators (i.e. the irreducible string operators are labeled by the representation of $D(G)$). But I'm not sure if the pictorial way of understanding quasiparticle creations still hold. Namely, If i were to create a pair of quasiparticles (labeled by rep $\alpha$ of $D(G)$) at points $a$ and $b$, is it the same as laying a type $\alpha$ string along a path connecting $a$ and $b$? Jan 14 '16 at 7:14
• I think the tricky part is if we take the R symbols in a quantum double as given, how do we obtain the $\Omega$ symbols (c.f. eq.(51)). Because in order to make sense of the above pictorial definition of string operators, one needs to resolve the crossings between strings labeled by irrep of $D(G)$, and the original strings in the string-net condensate, which doesn't look obvious to me. Jan 14 '16 at 7:55
• Also, if we go beyond quantum double models, say the double semion model, are the string operators also labeled by irreps of some algebraic object? I suppose it has something to do with braided fusion categories (and in the above case, the semion category). Jan 14 '16 at 10:55

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.)
• Thanks Zoltan! It is very helpful. I'm still confused about the Drinfeld center though. Let's say we take the input fusion category $C$ to be $\textbf{Rep}G$, where $G$ is a finite group. What is $Z(C)$ in this case? Is it the quantum double $D(G)$ of G? Also, I think the quasiparticles (and irreducible string operators) should be labeled by irreps of $D(G)$. Dominic and you mentioned that in the more general setting, it is the simple objects in $Z(C)$ that labels the quasiparticles. How should I think of irreps of $D(G)$ in terms of the simple objects language? Jan 15 '16 at 23:40
• Also, for finite groups I think it is possible to define string operators by an algorithm based on Fourier transform as you mentioned. But if we go beyond the $C = \textbf{Rep}G$ case, where we cannot map the model to a quantum double model. How do we obtain the $\Omega$ symbols? page 14 of arxiv.org/abs/cond-mat/0404617 gave an example where $C$ is the Fibonacci category. I'm not sure how they obtained the $\Omega$ symbols in eq.(51). Does one have to solve eq.(22) by brute force? Jan 15 '16 at 23:59