content_image/309392.jpg
by changing coordinates to U = r/α ′ the α ′ dependence drops out of S$_{D}$$_{3}$ . The controlling parameter for Eq. (2.42) then turns out to be g$_{s}$N ( ∂U ) $^{2}$/U 4 [322]. The lowest order term from the action yields N = 4 SU ( N ) super-Yang-Mills from the F$_{αβ}$ term. The quadratic term has no quantum corrections and the quartic term has only a one-loop correction, consistent with N = 4 SYM [326, 327]. This loop correction has been calculated from the gauge theory and string theory, and the two agree [328]. Moreover, it can be argued that all higher order terms in Eq. (2.42) are determined from the fourth-order term [312]. In order for this series to make sense g$_{s}$N ( ∂U ) $^{2}$/U 4 ≪ 1. In particular in the supergravity regime of the AdS/CFT duality g$_{s}$N ≫ 1, and the higher order terms beyond SU ( N ) may become important. 

## 2.3.3 Motivation for the AdS/CFT Correspondence 

In this section we will follow [322] closely for a schematic motivation of the AdS/CFT conjecture (see the same for a very good review of the correspon- dence as well as an extensive bibliography). Consider from a field theoretic standpoint Type IIB string theory in a 10 dimensional spacetime with a stack of N coincident extremal ( r$_{+}$ = r$_{−}$ ≡ R ) D 3 branes at the origin. At ener- gies small compared to the string scale 1 /l$_{s}$ = 1 / √ α ′ only the massless string modes can be excited. The full low energy effective action is then 

<img src='content_image/309392.jpg'>

