by $SO(4,2) \times
SO(6)$, though if we include fermions, the full supersymmetric isometry group is [$PSU(2,2|4)$]{}. Note that this is exactly the same as the full global symmetry group of the low energy limit of the open string sector ([*i.e.*]{} SYM theory).

We see that the radius $L$, of both the AdS throat and the $S^5$, in string units is given in terms of the gauge theory parameters as

$$L^4= g^2_{YM}\, N_c\, {\alpha^\prime}^{2}=\lambda{\alpha'}^2 \,.$$

Therefore, in order that the stringy modes be unimportant, $L\gg\sqrt{\alpha'}$, which translates into gauge theory language as $\lambda = g^2_{YM}\, N_c \gg 1$.

$\cal N=$ 4 Super-Yang-Mills theory
-----------------------------------

$\cal N =$ 4 $SU(N)$ supersymmetric Yang-Mills theory (SYM) in four dimensions (the dimensionality of the world-volume of the D3-branes) has one vector field, $A_\mu$, six scalars fields $\phi^I$ ($I=1...6$), and four fermions $\chi_{\alpha}^i$, $\chi_{\dot\alpha}^{\bar i}$ ( $i,\bar i=1,2,3,4$) which are in the ${\bf 4}$ and $\bar {\bf 4}$ representations of the $SU(4) = SO(6)$ $R$-symmetry group.

This theory naturally arises on the surface of a D3 brane in type IIB superstring theory. Open strings generate a massless gauge field in ten dimensions. When the open string ends are restricted to a 3+1 dimensional subspace the ten 