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## 2.3 Matching the Symmetries 

What symmetries do each side of the description have? On the AdS$_{5}$ × S 5 side we clearly have two symmetry groups: the SO (6) of the 5-sphere, and the symmetry of the AdS$_{5}$ spacetime is SO(4,2) [22]. What does this relate to on the super Yang-Mills side? Since it is N = 4 there are four gauginos, hence there is an SU (4)$_{R}$ ∼ = SO (6) symmetry. It is also conformal in all four dimensions: the conformal group in four dimensions has a symmetry group SO(4,2). So we can identify corresponding symmetries on either side of the correspondence: 

## 2.4 Regime of usefulness 

<img src='content_image/301476.jpg'>

In what regime are our approximations valid? On the supergravity side, the two regimes of interest decoupled when the wavelength of the supergravity excitations were much larger than the string length: 

<img src='content_image/301477.jpg'>

But on the field theory side we need the ’t Hooft coupling to be much less than one if the theory is to be in the perturbative regime: 

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Clearly the two inequalities are incompatible, which leads us to the conclu- sion that when the gravity dual is weakly coupled, the field theory must be 

