a matrix model defined by the partition function $$Z_{TEK}=\int\prod_{\mu=1}^4 dU_{\mu}\exp(-S_{TEK}),$$ with the action $$S_{TEK}=-\beta N\sum_{\mu\neq\nu}Z_{\mu\nu}{\rm Tr}
~U_\mu U_\nu U_\mu^\dagger U_\nu^\dagger,
\label{EQ:TEK_action}$$ where $U_{\mu}$ and $dU_\mu\ (\mu=1,2,3,4)$ are link variables and Haar measure. The phase factors $Z_{\mu\nu}$ are $$Z_{\mu\nu}=\exp\left(2\pi i n_{\mu\nu}/N\right),\qquad
n_{\mu\nu}=-n_{\nu\mu}\in \mathbb{Z}_N.$$ The Wilson loop operator also contains the phase $Z({\cal C})$ as $$W_{TEK}({\cal C})\equiv Z({\cal C})\langle\hat{W}({\cal C})\rangle,$$ where $\hat{W}({\cal C})$ is the trace of the product of link variables along a contour $\cal C$ and $Z({\cal C})$ is the product of $Z_{\mu\nu}$’s which correspond to the plaquettes in a surface whose boundary is $\cal C$. This model is obtained by dimensional reduction of the Wilson’s lattice gauge theory with the twisted boundary condition. With these definitions, the loop equations in the TEK model take the same form as those in the ordinary lattice gauge theory if the $\mathbb{Z}_N^4$ symmetry, which we discuss in section \[SEC:U(1)symmetry\], is not broken.

Twist prescriptions and classical solutions {#SEC:twist prescription}
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In the weak coupling limit, the path-integral is dominated by the configuration which gives the minimum to the action. This configuration $U^{(0)}_\mu=\Gamma_{\mu}$ satisfies the ’t Hooft algebra $$\Gamma_{\mu}\Gamma_{\nu}=Z_{\nu\mu}\Gamma_{\nu}\Gamma_{\mu},$$ and is called “twist-eater”. The most popular twist might be the minimal symmetric twist (standard 