61 Chapter 2: Motivation S$_{bulk}$ is the 10 D supergravity action with higher derivative corrections. S$_{branes}$ is the 3 + 1 D SYM action with higher derivative corrections. Finally S$_{int}$ describes the interaction between the brane and bulk modes. It turns out that all of these actions are controlled by α $^{′}$: the bulk action is controlled by Newton’s constant κ , the brane action by α ′ $^{2}$, and S$_{int}$ also by κ . Then taking the low energy limit κ ∼ g$_{s}$α ′ 2 → 0 (specifically α ′ → 0 with U = r/α ′ fixed), the higher derivative and interaction terms all drop out. We are left with noninteracting supergravity in the bulk and N = 4 SU ( N ) SYM on the branes, which are decoupled from each other. On the other hand consider the geometrical interpretation of the low energy limit of Type IIB string theory in 10 dimensions. From Eq. (2.38) we have that There are two regions of importance: the near horizon limit, r → 0, and the asymptotically flat limit, r → ∞ . In the flat limit the metric is the usual Minkowski one corresponding to, because of α ′ → 0, free supergravity. In the low energy limit these massless excitations decouple from the brane physics as their cross section goes ∼ ω $^{3}$R8 [329] (note that cross sections in n spatial dimensions for problems with p translational symmetries have dimension L n − 1 − $^{p}$). On the other hand the nontrivial g$_{tt}$ component of Eq. ds 2 = f − 1 / $^{2}$( − dt 2 + dx 2 1 + dx 2 2 + dx 2 $_{3}$) + f 1 / $^{2}$( dr 2 + r $^{2}$d Ω 2 $_{5}$) , (2.45) f = 1 + R 4 r 4 , R 4 = 4 πg$_{s}$l 4 $_{s}$N. (2.46)