& & & \multirow{2}{0mm}[-1mm]{\it\huge O}         \\
 & e^{2\pi i/L} &                     &        &                   \\
 &              & e^{2\pi i\cdot 2/L} &        &                   \\
 &              &                     & \ddots &                   \\
\multirow{2}{0mm}[4mm]{\it\huge O} & & &        & e^{2\pi i(L-1)/L}
\end{array}
\right),\end{aligned}$$ which satisfy the little ’t Hooft algebra $$\hat{C}_L\hat{S}_L =  e^{-2\pi i/L}\hat{S}_L\hat{C}_L.$$ Using these matrices, the twist-eater configuration for the skew-diagonal form (\[EQ:minimal\_skew\_form\]) is easily constructed as $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\hat{C}_L     \otimes \mathbbm{1}_L, \quad &
\Gamma_2=\hat{S}_L     \otimes \mathbbm{1}_L,       \\
\Gamma_3=\mathbbm{1}_L \otimes \hat{C}_L,           &
\Gamma_4=\mathbbm{1}_L \otimes \hat{S}_L.
\end{array}\end{aligned}$$ From (\[EQ:minimal\_skew\_form\]) we can also construct the twist-eater configuration for the minimal symmetric twist (\[EQ:minimal\_sym\_form\]) as $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\quad\;\hat{C}_L   \otimes \mathbbm{1}_L, \quad &
\Gamma_2=\hat{S}_L\hat{C}_L \otimes \hat{C}_L,           \\
\Gamma_3=\hat{S}_L\hat{C}_L \otimes \hat{S}_L,           &
\Gamma_4=\quad\;\hat{S}_L   \otimes \mathbbm{1}_L. 
\end{array}\end{aligned}$$ Although these forms are different only by the coordinate transformation, they can give different results except the weak coupling limit as seen in next section.

Another kind of the twist we consider in this article is $$\begin{aligned}
n_{\mu\nu}=
\left(\begin{array}{cc|cc}
 0  &  mL &  0  & 0 \\
-mL &  0  &  0  & 0 \\
\hline
 0  &  0  &  0  & mL\\
 0  &  0  & -mL & 0
\end{array}
\right),\qquad N=mL^2
\label{EQ:generic_twist}\end{aligned}$$ with classical solution $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\hat{C}_L     \otimes \mathbbm{1}_L \otimes 