63 Chapter 2: Motivation Perturbative field theory is a consistent approach for calculations when where the final relation came from Eq. (2.46). Contrariwise string theory is tractable in the classical supergravity limit, in which case the characteristic length scale of the problem, R , is large compared to the two length scales of quantum string theory: the string length, l$_{s}$ , and the Planck length, l$_{p}$ ; this ensures no string corrections and no loop corrections, respectively. Demanding this implies g 2 $_{SY M}$N = 4 πg$_{s}$N = R 4 l 4 s ≪ 1 , (2.48) R 4 l 4 s = 4 πg$_{s}$N = g 2 $_{SY M}$N ≫ 1; R 4 l 4 p ∼ N ≫ 1 . (2.49) There is an extra subtlety due to the SL (2 ,$_{Z}$ ) self-duality symmetry of both the gauge and string theory. For the coupling τ = 4 πi/g 2 SY M = i/g$_{s}$ the theories are invariant under τ → ( aτ + b ) / ( cτ + d ) where ad − bc = 1. We can see that one can take g$_{s}$ → 1 /g$_{s}$ with a = d = 0 and b = 1, c = − 1; a strong string coupling can be swapped for a weak one and vice-versa, but the number of colors must always remain large. This is the origin of the λ large and fixed while N → ∞ limit; this guarantees an S transformation leaves R large compared to both l$_{s}$ and l$_{p}$ . Now we will show that the near horizon limit that is the background for the dual string theory is AdS$_{5}$ × S $^{5}$. To see this, taking r → 0 in Eq. (2.45)