the vertices of a cubic lattice and ${\mathbb{Z}}_2$ gauge variables $A_{ij}$ on the links, and choose the following energy $$\begin{aligned}
  \label{eq:E_3D}
  -\beta\mathscr{H}
  = J_1 \sum_{p} (AAAA)_p + J_2 \sum_{p} (-)^{\omega_p} (AAAA)_p\, ,\end{aligned}$$ with $(AAAA)_p$ being the product of the four $A_{ij}$ surrounding the plaquette $p$.

In analogy with our $2D$ analysis we would now want to perform some non-local transformation to decouple the gauge variables from the spins. Even though both flux and vorticity lines form closed polygons, the number of distinct flux configurations, which spans all such polygons, is bigger than that of vorticity configurations which only span a subset. Therefore, for any vorticity there exists a matching flux although the converse is not true. It follows that there exists $A_{\sigma}$ such that $(A_{\sigma}A_{\sigma}A_{\sigma}A_{\sigma})_p =
(-)^{\omega_p}$. Defining $\tilde{A} = A A_{\sigma}$, we obtain $$\begin{aligned}
   \label{eq:Atilde}
  -\beta\mathscr{H}
 &= J_1 \sum_{p} (-)^{\omega_p} (\tilde{A}\tilde{A}\tilde{A}\tilde{A})_p + J_2 \sum_{p} (\tilde{A}\tilde{A}\tilde{A}\tilde{A})_p \, .\end{aligned}$$

There are two points in phase space where the gauge and spin degrees of freedom decouple. The trivial case is $J_2 = 0$ which implies free $\sigma$’s and a standard ${\mathbb{Z}}_2$ gauge theory for the $A$’s. For $J_1>J_c$, where $J_c=0.762(2)$ is the critical temperature of the dual Ising model on the cubic 