List of Figures 1.1 Plot of the solution to the first loop beta function for QCD . 11 1.2 Two examples of contributions to the vacuum amplitude in double line notation. The diagram on the left is planar and hence contributes N $^{2}$. The diagram on the right is of genus one and hence contributes N 0 . . . . . . . . . . . . . . . . . 14 1.3 Possible forms of the β function with non-trivial zeros: (a) ultraviolet-stable fixed point; (b) infrared-stable fixed point . 15 1.4 The vector current two-point function, Π$_{V}$ ( − q $^{2}$). The grey blob represents all possible one particle irreducible diagrams that can be inserted. . . . . . . . . . . . . . . . . . . . . . . 21 1.5 One contribution to the vector current two-point function . . 22 1.6 One of the diagrams that leads to the axial vector anomaly in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 If we place the D7 brane(s) at any non-zero distance from the D3 branes, we break the conformal symmetry . . . . . . 47 3.2 D7-brane flows for m = 1 with several different condensate values in AdS$_{5}$ × S 5 background. Only the c = 0 solution allows a consistent theory . . . . . . . . . . . . . . . . . . . 51 3.3 Numerical calculation of the meson spectrum in AdS$_{5}$ × S $^{5}$. The dots are the analytical values. . . . . . . . . . . . . . . 55 3.4 Plot of three solutions around the M = √$_{2 solution. . . . . .}$ 56 4.1 Wilson loop used to calculate the quark-antiquark force. The vertical direction represents Euclidean time. The horizontal direction indicates one of the spatial coordinates . . . . . . . 61 4.2 Solutions for the ω$_{6}$ flow when ω$_{5}$ = 0 showing the depen- dence of the condensate on the quark mass . . . . . . . . . . 67 v