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My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.

Elementary excitations in the WW model can be point particles, loop defects and more general defects (see page 12 of arXiv:1104.2632). It is stated their description is closely related to the boundary condition of 3-manifolds.

From my understanding, a boundary condition is a collection of ribbon ends on the surface boundary Y. One can then define a 1-category A(Y) in which objects are the boundary conditions, morphisms are linear combinations of string-nets in Y x [0,1], between Yx0 and Yx1, and composition is given by 'stacking'. If I am not mistaken, excitations are then associated with the hom spaces of this category.

Does this construction encapsulate all types of higher defects including loop defects? If so, then how can I think of loop defects in this setting?

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In the original WW paper, we distinguish between "crude" and "topological" boundary conditions. (I think "algebraic" would be a better name than "topological" here, but perhaps it is too late to change the naming conventions. Or perhaps not.)

As you write, we associate a linear category $A(Y)$ to a 2-manifold $Y$. The crude boundary conditions are the objects of $A(Y)$. The topological/algebraic boundary conditions are the representations of $A(Y)$. An elementary excitation is an irreducible representation of $A(Y)$. (In this case $Y$ is the boundary of the excitation -- a sphere in the case of a point excitation, a torus in the case of a loop excitation.)

The boundary of a loop excitation is a torus $T$, so the possible elementary loop excitations correspond to irreducible representations of $A(T)$.

Defects are another sort of boundary condition, and correspond to higher categorical representations. For example, a domain wall type defect between a WW model and the vacuum corresponds to a (higher) representation of the 3-category $A(p)$ associated to a point $p$.

Codimension-2 defects (e.g. loop defects) correspond to representations of the 2-category $A(S)$, where $S$ is a circle. (You should think of $S$ as a small circle which links the loop defect, not the large circle (loop) which describes the location of the loop defect.) For a WW model, the 0-morphisms of $A(S)$ are trivial (only one 0-morphism), the 1-morphisms are ribbon end points in $S\times I$, and the 2-morphisms are linear combinations of string nets in $S\times D$, modulo the usual relations. (Here $D$ is a disk.) Since the 0-morphisms are trivial, this 2-category can be thought of as a tensor category.

In this context, a representation of $A(S)$ is just a module category for $A(S)$.

Every loop defect gives rise to a loop excitation, but not vice-versa.

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  • $\begingroup$ Thanks for the answer. For someone familiar with the Hamiltonian version of the WW and LW models, and trying to understand the skein space/string-net picture, is there an intuitive way to think about the various ingredients involved including quasiparticle excitations? This is not obvious to me, for example, the term boundary, as in lattice boundary and manifold boundary is not the same. $\endgroup$ – Abbas Mar 30 '15 at 15:23
  • $\begingroup$ I tend to think it terms of string nets, so I'm not sure I can say what is intuitive from the lattice hamiltonian point of view, but I'll try. Crude boundary conditions correspond to fixing certain spins near the boundary of the lattice, I think. Defects can be described nicely from the lattice point of view. Along the defect there are special degrees of freedom (spins), and near the defect there are special terms in the hamiltonian describing the interaction between the defect spins and the bulk spins. ... $\endgroup$ – Kevin Walker Mar 30 '15 at 17:23
  • $\begingroup$ ... For example, a module category for $A(S)$ can be used to modify the WW hamiltonian near a loop, and thus describe a loop defect. For excitations, I'm not sure what counts as "intuitive". If you have a favorite way of understanding excitations in, say, the toric code, then probably there is an analogous description of excitations for LW and WW models. $\endgroup$ – Kevin Walker Mar 30 '15 at 17:27

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