How I can get the numerical factor in the relation between string coupling and YM coupling? I'm trying to understand some references about Wilson loops being used to test AdS/CFT. Some of them are


*

*Nadav Drukker, David J. Gross: An Exact Prediction of N=4 SUSYM Theory for String Theory

*Gordon W. Semenoff: Perturbative computations in SUSYM: Testing AdS/CFT 
It seems that they use
$$g_{ym}^2=4\pi g_s \tag{A}$$
But, it is not clear for me how to get that identification. I tried to expand the D3-brane action and compare to the Yang-Mills action. This is done for example in


*

*Richard J. Szabo: BUSSTEPP Lectures on String Theory

*Ammon, Martin; Erdmenger, Johanna; Gauge-gravity duality - foundations and applications-Cambridge University Press (2015)
but I got $g_{ym}^2=2\pi g_s$.
Is (A) correct? How can I get that relation?
 A: Both are correct, but depends on the use of normalisation for the Lie algebra generators when writing the Yang-Mills action.
Recall that a Lie algebra valued field strength is written
\begin{equation}
F_{\mu\nu}=F_{\mu\nu}^{a}T^a
\end{equation}
and you choose a normalisation constant, $c$, for the Lie algebra generators in the fundamental representation:
\begin{equation}
\mathrm{Tr}\left[T^a,T^b\right]=c\delta^{ab}.
\end{equation}
Deriving the D3-brane Yang-Mills action from the lowest DBI action gives an action of the form
\begin{equation}
S_{\mathrm{YM}}=-\frac{1}{4(2\pi g_s)}\int \mathrm{d}^4x\,\mathrm{Tr}\left[F_{\mu\nu}F^{\mu\nu}\right]=-\frac{c}{4(2\pi g_s)}\int \mathrm{d}^4x\,F_{\mu\nu}^a F^{a,\mu\nu},
\end{equation}
and with canonical normalisation we therefore have that
\begin{equation}
g_{\mathrm{YM}}^2=\frac{2\pi g_s}{c}.
\end{equation}
The most often used choice is $c=1/2$ giving the relation you referenced in the papers:
\begin{equation}
g_{\mathrm{YM}}^2=4\pi g_s.
\end{equation}
