theories on $m$-coincident four-dimensional fuzzy tori.

$\mathbb{Z}_N^4$ symmetry {#SEC:U(1)symmetry}
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The $\mathbb{Z}_N^4$ symmetry plays a crucial role in the Eguchi-Kawai equivalence. Generally, the YM theory with a periodic boundary condition has a critical size. If we shrink the volume of the system beyond the critical size, we encounter the center symmetry breaking, which is just the same as the finite temperature system. In the EK model, which is a single hyper-cubic model, the critical size corresponds to $\beta_c\sim 0.19$ in the lattice coupling. In the region less than the $\beta_c$ – the strong coupling region – the center symmetry ${\mathbb Z}_N^4$ is maintained. On the other hand, in the region larger than $\beta_c$ – the weak coupling region – the symmetry is spontaneously broken, and then the EK equivalence does not hold.

The TEK model avoids this problem by imposing the twisted boundary condition on the system instead of the periodic one. In the weak coupling limit the path integral is dominated by the vacuum configuration, which is twist-eater configurations, as we already mentioned. These configurations are invariant under global ${\mathbb Z}_L^4$ transformation $$U_\mu\to e^{i\theta_\mu}U_\mu, \qquad e^{i\theta_\mu}\in {\mathbb Z}_L,$$ which is regarded as the $U(1)^4$ symmetry in the large-$N$ limit. 