color branes, $N_c$.

To avoid singularity, the coordinate $\tau$ satisfies the following periodic condition, $$\begin{aligned}
\tau \sim \tau + \delta \tau\,\,,\qquad\qquad \delta\tau=\frac{4\pi
R^2}{5 U_{KK}}\,\,.\end{aligned}$$ Also, the Kaluza-Klein mass scale of this compact dimension is $$\begin{aligned}
M_{KK}=\frac{2\pi}{\delta\tau}= \frac{5}{2} \frac{ \ U_{KK}}{R^2},\end{aligned}$$ and dual gauge field theory for this background is non supersymmetric. Also, the Yang-Mills coupling constants can be defined as a function of string theory parameters using the DBI action as follows $$\begin{aligned}
g_{YM}^2=\frac{g_s}{\mu_4 \,(2\pi \alpha')^2\, \delta \tau},\end{aligned}$$ where $\alpha'=l_s^2$ is the Regge slope parameter and $l_s$ is the string length. Also, the ’t Hooft coupling is $\lambda=g_{YM}^2\,
N_c$.

meson sector
------------

In AdS/QCD, there is a gauge field living in the bulk AdS whose dynamics is dual to the meson sector of QCD such as pions and higher resonances. The gauge field on the $D4$ brane includes five components, $A_\mu (\mu=0,1,2,3)$ and $A_U$. The $D4$ brane action is given by \[49\] $$\begin{aligned}
S_{D4}= -\mu_4 \int d^5 x e^{-\phi} \sqrt{-\det(g_{MN}+2\pi\alpha'
F_{MN})}+S_{CS}, \label{FF}\end{aligned}$$ where $F_{MN}=\partial_M A_N-\partial_N A_M-i[A_M,A_N],$ $(M,N=0,1,..5)$ is the field strength tensor, and the $A_M$ is the $U(N_f)$ gauge field on the $D4$ brane. The second term in the above action is the Chern-Simons action and $\mu_4=2\pi / (2\pi ls)^5$. 