2.3 Matching the Symmetries 2.4 Regime of usefulness 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: 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: 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: 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 N = 4 SYM AdS$_{5}$ × S 5 SO(4,2) conformal group spacetime symmetry SU(4) ∼ =SO(6) R-symmetry S 5 isometry 1 ≪ R 4 l 4 s ∼ g 2 $_{YM}$N ∼ g$_{s}$N (2.6) 1 ≫ g 2 $_{YM}$N ∼ g$_{s}$N ∼ R 4 l 4 s (2.7) 39