In the usual formulation of lattice gauge theories, one considers gauge variables on the links of a lattice (often hypercubic) taking value in some representation of a gauge group, $U_{ij} \in G$. The action can generally formulated as the sum of Wilson loops around plaquettes, $$S_p = \beta \sum_{\square}\mathrm{Tr}\left(\prod_{(ij)\in \square} U_{ij}\right)$$ Gauge invariance in such a model is manifest in the gauge invariance of the Wilson loop under the transformation of link variables $U_{ij} \rightarrow g_i^{-1} U_{ij} g_{j}$, where the $g_i$ are arbitrary elements of $G$ defined on the sites of the lattice. The link variables are then thought of as encoding the gauge field according to $$U_{ij} = \exp(\mathrm{i}\, e\!\int_i^j A_\mu \,dx^\mu)$$ and one can generally show that in the continuum limit the above action would recover the Yang Mills action, $S_p \sim \int \mathrm{d}^{d}x F_{\mu\nu}F^{\mu\nu}$.

On the other hand, vertex models are generally also formulated in terms of variables living on the bonds of a lattice, but instead of having an action defined on plaquettes, we define it with respect to the "stars", for example, we could have something like $$S_v \sim \beta \sum_{i} \left(\sum_{j @i} \sigma_{ij} \right)^2$$ where $j@i$ mean "$j$ adjacent to $i$". The minimum of this action is clearly given by minimizing the "star" of every vertex of the lattice, i.e. it locally has zero "divergence". In this case I do not immediately see where the gauge theoretic constructions of a vector potential and Wilson loops or a field strength tensor come into play, I assume that the zero-divergence constraint means that we can write $\sigma \sim \mathrm{curl}(A)$ or something like that. The $\sigma_{ij}$ seem like they ought to be interpreted as "electric fluxes" rather Wilson links. There is also a relation here to spin models (the $\sigma_{ij}$ could be spin variables for example). Also, the above action $S_v$ should probably rather be interpreted as a Hamiltonian on a spatial lattice, whereas $S_p$ is defined on a lattice which includes timeline bonds (usually in Euclidean time, however, so I'm not sure how important the distinction is).

So, my question is, in what sense are vertex models/spin models (in some appropriate context) described by gauge theories? Does one need to introduce a dual lattice to make the connection explicit? I would appreciate any direction to dig into this topic further. I am especially interested in what happens when the "gauge variables" are discrete, like Ising spins or the Potts model.


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