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
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?