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?



Loop defects in Walker-Wang model
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.


