components of the gauge field naturally break into a 3+1 dimensional gauge field and 6 scalar fields. The fermionic super-partners naturally separate to complete the 3+1 dimensional super-multiplets.

The beta function of $\cal N=$ 4 SYM theory vanishes to all orders in perturbation theory, $\beta=0$. This implies the theory is conformal with conformal symmetry group $SO(4,2)$ also at the quantum level. Moreover this theory has a global $SU(4)$ R symmetry group. The complete superconformal group is $SU(2,2| 4)$, of which both $SO(4,2)$ and $SU(4)$ are bosonic subgroups.

The AdS/CFT Correspondence 
---------------------------

The AdS/CFT correspondence which was first suggested by Maldacena \[12\] in 1997, states that Type IIB string theory on $(AdS_5 ×
S^5)_N$ plus some appropriate boundary conditions (and possibly also some boundary degrees of freedom) is dual to $\cal N = $4, d = 3 + 1 U(N) super-Yang-Mills. There are three different versions of this conjecture \[63\], depending on the precise form of the limits taken. In the strong version, Type IIB string theory on $AdS_5 \times
S^5$ is dual to $SU(N_c)$ SYM theory. The mild version relates Classical type IIB strings on $AdS_5 \times
S^5$ to planar $SU(N_c)$ SYM theory. But the mostly adopted form of the 