multiple of $|\mathbb{Z}_2|=2$. This is analogous to what one does when calculating vorticity of a U$(1)$ variable ($\phi$) where derivatives ($i \phi^{-1} \partial_l \phi$) are taken in U$(1)$ but then integrated over as elements in $\mathbb{R}$ whose sum can now be a non-zero multiple of $2\pi$.

In analogy with usual vorticity, the discrete vorticity obeys a discrete version of the zero divergence constraint: Given any box on the square lattice, $\sum_{p\in\text{box}}\omega_p = 0\mod 2$. This can be shown by noting that for each box we can choose a clockwise orientation (when looking from inside the box) on each plaquette. Consequently, each link on the box would appear exactly twice with opposite values of $\epsilon_{ij}^p$. Therefore discrete vorticity lines form polygons on the dual lattice which obey the exact same branching rules as fluxes in a ${\mathbb{Z}}_2$ gauge theory.

Tools from lattice gauge theory, specifically cellular and simplicial cohomology, shed further light on this quantity. A thorough discussion of these aspects are relegated below in section \[sec:cohom\] where they will be used to define discrete vorticity for other abelian groups. Armed with the notion of discrete vorticity and its properties, we can now introduce the $3D$ model. Consider spins $\sigma_i$ on 