the “noncommutative coordinate” $\hat{x}^\mu=\left(\theta^{-1}\right)^{\mu\nu}\hat{p}_\nu$. Then we have $$[\hat{x}^\mu,\hat{x}^\nu]=-i(\theta^{-1})^{\mu\nu}.$$ This commutation relation is the same as that of coordinate on fuzzy ${\mathbb R}^4$ with noncommutativity parameter $\theta$, and hence functions of $\hat{x}$ can be mapped to functions on fuzzy ${\mathbb R}^4$. More precisely, we have the following mapping rule: $$\begin{aligned}
\begin{array}{ccc}
f(\hat{x})=\sum_k\tilde{f}(k)e^{ik\hat{x}} & \leftrightarrow &
f(x)=\sum_k\tilde{f}(k)e^{ikx},\\
f(\hat{x})g(\hat{x})                       & \leftrightarrow &
f(x)\star g(x),\\
i[\hat{p}_{\mu},\ \cdot\ ]                 & \leftrightarrow &
\partial_{\mu},\\
{\rm Tr}                                   & \leftrightarrow &
\frac{\sqrt{\det\theta}}{4\pi^2}\int d^4x,  
\end{array}\end{aligned}$$ where $\star$ represents the noncommutative star product, $$f(x)\star g(x)=
f(x)\exp\left(-\frac{i}{2}\overset{\leftarrow}{\partial}_{\mu}
(\theta^{-1})^{\mu\nu}\overset{\rightarrow}{\partial}_{\nu}\right) g(x),$$ and we obtain $U(1)$ NCYM action $$S_{U(1)NC}=-\frac{1}{4g_{NC}^2}\int d^4x ~F_{\mu\nu}\star F_{\mu\nu}$$ with coupling constant $$g_{NC}^2=4\pi^2g^2/\sqrt{\det\theta}.$$ In the same way, by expanding the action (\[EQ:continuum\_TEK\]) around $A_\mu^{(0)}=\hat{p}_\mu\otimes\mathbbm{1}_m$, $U(m)$ NCYM can be obtained. From (\[EQ:generic\_twist-eater\]), it is apparent that the generic twist gives the $U(m)$ NCYM. Intuitively, the vacuum configuration (\[EQ:generic\_twist-eater\]) describes $m$-coincident fuzzy tori and fuzzy ${\mathbb R}^4$ is realized as a tangent space.

In order to keep the noncommutative scale $\theta$ finite, we should take the double scaling limit with $$\begin{aligned}
a^{-1}\sim\sqrt{L}\sim N^{1/4}. \end{aligned}$$ One-loop beta function for $U(m)$ NCYM is given by [@Minwalla:1999px][^3] $$\begin{aligned}
\beta_{\rm 1-loop}(g_{NC})=
-\frac{2}{(4\pi)^2}\frac{11}{3}mg_{NC}^3+O(g_{NC}^5). \end{aligned}$$ Therefore, the ’t Hooft coupling $\beta$ scales as $$\begin{aligned}
\beta \sim \frac{1}{g_{NC}^2} \sim \log 