# Dualities in 2+1D lattice gauge theories

A nice way to understand $\mathbb{Z}_2$ gauge theories is via duality transformations. For example, it is illustrated in http://arxiv.org/abs/1202.3120 that a $\mathbb{Z}_2$ gauge theory (with Ising matter fields) obtained by gauging the $\mathbb{Z}_2$ symmetry of a spin model can be dualized to a $\mathbb{Z}_2$ string-net model on the dual lattice. The explicit duality mapping constructed there is $\tau_l^z = \sigma_p^z \mu_{pq}^z \sigma_q^z$, $\tau_l^x = \mu_{pq}^x$, where $\sigma_p$ ($\mu_{pq}$) is the Ising matter fields (gauge fields) living on the vertices (links) of the original lattice, and $\tau_l$ is the dual $\mathbb{Z}_2$ gauge fields living on the links of the dual lattice. A particular nice thing about the above duality is that matter fields are gone in the dual picture, and the ground state consists just of closed electric field lines, which is simple and one can easily extract the F symbols for the theory and so on.

I want to generalize the above duality to arbitrary (finite) group $G$. The way to gauge a many body quantum state $\vert \Psi \rangle$ with global symmetry $G$ is to first enlarge the total Hilbert space to include the gauge-field degrees of freedom (labeled by group elements $g \in G$) living on the links. Then we tensor $\vert \Psi \rangle$ with the trivial gauge field configuration (so that the plaquette term in the gauged Hamiltonian is satisfied). Finally, we apply the Gauss'-law projector $P_v$ on each site $v$ (so that the Gauss' law constraint on each site $v$ is satisfied). More concretely,

$\vert \Psi_{\text{gauged}} \rangle = \prod_{v\in \Gamma} P_v (\vert \Psi \rangle \otimes_{e \in \Gamma}\vert 1 \rangle_e)= \sum_{\{g_v \}} \Big(\prod_{v\in \Gamma} U_v(g_v) \vert \Psi \rangle \Big)\otimes_{e \in \Gamma}\vert g_{v_{e^-}}g_{v_{e^+}}^{-1} \rangle_e$.

The dual description of the above $G$-gauge theory with matter fields may be a string-net state with gauge group $G$. The complication in the general-$G$ case is that the vertex degree of freedom are labeled by unitary representations of $G$ (in the $\mathbb{Z}_2$ case the vertex degrees of freedom are just labeled by $\mathbb{Z}_2$ charges). Moreover, the dimension of the representation may not match the dimension of the gauge-field Hilbert space, so that one should expect some residual degree of freedom on the vertices even after applying the duality (Probably string-net with multiplicities?).