analogy to the last point, we let $\Lambda^2$ be a choice of orientation in $\wt\Lambda^2$.

-   If we need to identify the elements of $\Lambda^i$ with subsets of $\R^3$, we will make the natural identifications, e.g. a link $\ell=(x,y)\in\Lambda^1$ is the undirected closed line segment from $x$ to $y$.

-   We also need to consider subsets of the lattice, so given a connected subgraph $S\subset(\Lambda^0, \Lambda^1)$, we let $\Lambda_S^0$ be all the vertices in $S$, $\Lambda_S^1\subset\Lambda^1$ is the set of links which are edges in $S$, and $\Lambda_S^2\subset\Lambda^2$ is the set of those plaquettes whose sides are all in $S$.

We recall the lattice approximation on $\Lambda$ for a classical matter field with a classical gauge connection field acting on it. Fix a connected, compact Lie group $G$ (the gauge structure group), and let ${\big(\Cn,\,(\cdot,\cdot)_\Cn\big)}$ be a finite dimensional complex Hilbert space (the space of internal degrees of freedom of the matter field) on which $G$ acts smoothly as unitaries, so we take $G\subset U(\Cn)$. Then the classical matter fields are elements of $\prod\limits_{x\in\Lambda^0}\Cn$, on which the local gauge group $\prod\limits_{x\in \Lambda^0}G=G^{\Lambda^0}=\big\{\zeta:\Lambda^0\to G\big\}$ acts by pointwise multiplication. The classical gauge connections are maps $\Phi:\Lambda^1\to G$, i.e. elements of $\prod\limits_{\ell\in\Lambda^1}G$.

The 