# On string-like excitations in (3+1)d discrete gauge theory

(3+1)d discrete $$G$$-gauge theory (untwisted Dijkgraaf-Witten theory) has both point-like and loop-like excitations;

Point-like excitation is an electric charge labeled by an irreducible representation $$R_i$$ of $$G$$, which corresponds to an Wilson line operator, \begin{align} W_{R_i}(C):=\text{Tr}\left[R_i\left(\prod_{ij\in C}g_{ij}\right)\right]. \end{align} where $$C$$ is a closed line.

For loop-like excitations, first we can find a vortex line labeled by holonomy, which takes value in conjugacy class $$\chi$$ of $$G$$. A vortex line is created by an open surface operator, \begin{align} M_{\chi}(S):=\sum_{h\in\chi}\left(\prod_{ij\in S}B_{ij}(h)\right), \end{align} where $$S$$ is a surface on a dual lattice, and $$B_{ij}$$ is an operator which transforms a link variable $$g_{ij}\mapsto hg_{ij}$$. This operator implements a defect along $$S$$, and violates flatness at the boundary of $$S$$, if defined on an open surface.

We can think of attaching a charge to a vortex line, defined as an irreducible representation of $$G_\chi$$ (centralizer of $$\chi$$). For example, for $$S_3$$ gauge theory let us consider a vortex line labeled by $$\chi=((1,2,3), (1,3,2))$$. Then, we can define a charge of $$G_\chi=\mathbb{Z}_3$$ for the vortex line. Such a loop-like excitation associated with charge, should be generated by a composite object of open line and surface operator.

What is the explicit form of such operator corresponding to charged loop-like excitations, defined on discrete gauge theory on a lattice?