all $g\in G$. Using this, we define a unitary $W_\zeta:L^2(G)\to L^2(G)$ by $$(W_\zeta\varphi)(h):=\varphi(\zeta^{-1}\cdot h)=\varphi(\zeta(x_\ell)^{-1}\,h\,\zeta(y_\ell))$$ using the fact that $G$ is unimodular, where the inverse was introduced to ensure that $\zeta\to W_\zeta$ is a homomorphism. Note that $W_\zeta\psi_0=\psi_0$. So for the quantum observables ${U_G\cup T_{L^\infty(G)}}$, the gauge transformation becomes $$\label{GTfell}
T_f\mapsto W_\zeta T_f W_\zeta^{-1}=T_{W_\zeta f}\quad\hbox{and}\quad
U_g\mapsto W_\zeta U_g W_\zeta^{-1}=U_{\zeta(x_\ell)g\zeta(x_\ell)^{-1}}$$ for $f\in L^\infty(G)\subset L^2(G)$ and $g\in G$. Moreover each $W_\zeta$ preserves the space $C^\infty(G)$, hence Equation (\[GTfell\]) also implies that $$\label{PTfell}
W_\zeta P_X W_\zeta^{-1}=P\s{\zeta(x_\ell)X\zeta(x_\ell)^{-1}}.\quad\hbox{for}\quad
 X\in\mathfrak{g}.$$

Thus for the full system we define on $\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)$ the unitaries $$\label{LocalW}
{\widehat}{W}_\zeta:=U^F_\zeta\otimes\big(\bigotimes_{\ell\in\Lambda^1}W^{(\ell)}_\zeta\big),\quad
\zeta\in\gauc \Lambda$$ where $W^{(\ell)}_\zeta$ is the $W_\zeta$ operator on the $\ell^{\rm th}$ factor. Then the gauge transformation produced by $\zeta$ on the system of operators is given by ${\rm Ad}({\widehat}{W}_\zeta)$.

In particular, recalling $W_\zeta T\s\Phi_{ij}(\ell). W_\zeta^{-1}=T\s{W_\zeta \Phi_{ij}(\ell)}.$ we see that $$\begin{aligned}
\big(W_\zeta \Phi_{ij}(\ell)\big)(g)&=&\Phi_{ij}(\ell)(\zeta(x_\ell)^{-1}\,g\,\zeta(y_\ell))
=\big(e_i,\zeta(x_\ell)^{-1}\,g\,\zeta(y_\ell)e_j\big)\nonumber\\[1mm]
\label{PhiTfs}
&=&\sum_{n,m}[\zeta(x_\ell)^{-1}]_{in}\,\Phi_{nm}(\ell)(g)\,[\zeta(y_\ell)]_{mj}\end{aligned}$$ where $[\zeta(x_\ell)]_{in}=(e_i,\zeta(x_\ell)e_n)$ are the usual matrix elements, so it is clear that the indices of the quantum connection $T_{\Phi_{ij}(\ell)}$ transform correctly for the gauge transformation $\zeta^{-1}$.

Finally, we construct the appropriate field C\*-algebra for this model. For the fermion part, we already have the C\*-algebra ${\mathfrak F}_{\Lambda} =  {\rm CAR}\big(\prod\limits_{x\in\Lambda^0}\Cn)$. Fix a link $\ell$, hence a 