by adding N$_{f}$ probe D7 branes to the geometry. The exact dimensions occupied by the D7 and D3 branes are shown in table 4.2. We make a change of variables in the two directions perpendicular to both branes: where P indicates the pullback of the spacetime metric onto the D7 world volume. The pullback of a metric is defined as and introduce one flavour of quark via a D7 brane probe in the geometry. The Dirac Born Infeld (DBI) action for the probe is with x$_{0}$ → x$_{9}$ being the coordinates in the bulk, and ξ$_{0}$ → ξ$_{7}$ being the coordinates on the D7 brane. We choose the static gauge, which simply means x$_{i}$ = ξ$_{i}$ for i = 1 → 7, and x$_{8}$ = x$_{8}$ ( ξ$_{0}$, . . . , ξ$_{7}$ ) , x$_{9}$ = x$_{9}$ ( ξ$_{0}$, . . . , ξ$_{7}$ ). This action will determine how the D7 lies in the remaining r$_{5}$ − r$_{6}$ directions. Performing the necessary calculation we get 65 with ω 2 = ρ 2 + | Φ | 2 = ρ 2 + ω 2 5 + ω 2 $_{6}$. P[G$_{ab}$] = ∂x α ∂ξ a ∂x β ∂ξ b G$_{αβ}$ (4.20) S$_{D}$$_{7}$ = − 1 (2 π ) $^{7}$α ′ $^{4}$g$_{s}$ ∫ d $^{8}$ξe φ √ − det(P[G$_{ab}$]) (4.19) Φ = r$_{5}$ + ir$_{6}$ = σe iθ (4.18) S$_{D7}$ = − 1 (2 π ) $^{7}$α ′ $^{4}$g$_{s}$ ∫ d $^{8}$ξ ( ω 4 + b 4 ω 4 − b 4 ) Δ 2 ρ 3 ( ω 8 − b 8 ω 8 ) × √ 1 + | ∂$_{ρ}$ Φ | 2 + H ( ω 4 + b 4 ω 4 − b 4 ) 4 − δ 8 ω 4 − b 4 ω 4 | ∂$_{x}$ Φ | 2 (4.21)