low energy effective action which matches that of the corresponding low energy field theories. Here we are interested in checking the form predicted by the probe for the higher order corrections. The leading low energy effective action plus corrections for the bosons in the background described above, is obtained by expanding the self-dual action :

$T_3$ has no dependence on the string coupling constant. We obtain for the scalar terms obtained from the expansion of  after setting $y=0$:

Notice that each term in this action comes multiplied by a different power of $N$. As things stand, the $2n$ derivative term will come with a coefficient of $\tau_{2}^{n}f^{n-1}\sim N^{n-1}$. The full effective action for the probe interacting with $N$ coincident source threebranes, should come with an overall factor of $N$. This is achieved by noting that the coupling of the background, $\tau_{2}$ should be identified with

where $\tau_{SW}$ is the Seiberg-Witten effective coupling for the field theory of interest. With this identification, the string coupling is $O({1\over N})$ and explicitly goes to zero as $N\to\infty$. Notice also that $Ng_{s}=Ng_{YM}^{2}\sim\lambda$ so that the large ’t Hooft coupling limit corresponds to large $\lambda$. After making the $N$ dependence of $\tau_{2}$ explicit, we find 