(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?

Thanks in advance!


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