I'm trying to understand some references about Wilson loops being used to test AdS/CFT. Some of them are

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

but I got $g_{ym}^2=2\pi g_s$.

Is (A) correct? How can I get that relation?


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}

  • $\begingroup$ Are you sure? I did that actually for just one brane (N=1), in other words I obtained $U(1)$ Yang-Mills action. So there is no dependence of the choice of normalization of the Lie algebra generator, because we dont need even to define that. I think the dependence is in the definition of what we call $g_{YM}$ and not your constant $c$. $\endgroup$ – ConfusedPhDstudent Sep 22 '19 at 6:30
  • $\begingroup$ In the abelian case, there is of course no such constant and the relation is chosen so that you obtain the canonical normalisation, as you have calculated. The choice when it comes to non-abelian (as in AdS/CFT) is presented as I wrote it in "String Theory in a Nutshell" by Elias Kiritsis, section 13.4, page 412. $\endgroup$ – Sparticle Sep 23 '19 at 10:39

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