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Perturbative field theory is a consistent approach for calculations when 

where the final relation came from Eq. (2.46). 

<img src='content_image/325032.jpg'>

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 

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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) 

