= {g_{\rm YM}}^2 N/(4\pi^2)$ is the ’t Hooft coupling of $SU(N)$ ${\mathcal{N}}=4$ SYM.[^5] The energy $E$ of type IIB string states on $\mbox{AdS}_5\times S^5$ is identified with the energy $E$ of the dual ${\mathcal{N}}=4$ SYM states on ${\mathbb{R}}\times S^3$, or equivalently, with the scaling dimension of the dual operators of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}^4$. Similarly the angular momenta $J_{1,2,3}$ on $S^5$ for string states are identified with the three R-charges $J_{1,2,3}$ for states/operators of ${\mathcal{N}}=4$ SYM. Moreover the angular momenta $S_{1,2}$ for strings on $\mbox{AdS}_5$ are identified with the Cartan generators for the $SO(4)$ symmetry of the $S^3$ for the dual ${\mathcal{N}}=4$ SYM states on ${\mathbb{R}}\times S^3$, or equivalently, the $SO(4)$ symmetry of the ${\mathbb{R}}^4$ for the dual operators of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}^4$.

The string theory that we are interested in is type IIB string theory on $\mbox{AdS}_5 \times S^5$. The metric for this background is given by $$\label{adsmet}
ds^2 = R^2 \left[ - \cosh^2 \rho dt^2 + d\rho^2 +
\sinh^2 \rho d{\Omega'_3}^2 + d\theta^2 + \sin^2 \theta d\alpha^2
+ \cos^2 \theta d\Omega_3^2 \right]\, ,$$ with the five-form Ramond-Ramond field strength $$\label{adsF5}
F_{(5)} = 2 R^4 ( \cosh \rho \sinh^3 \rho dt d\rho d\Omega_3' +
 \sin \theta \cos^3 \theta 