How I can get the numerical factor in the relation between string coupling and YM coupling?

Question

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?

Answer 1

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}