content_image/303502.jpg content_image/303501.jpg content_image/303500.jpg content_image/303503.jpg
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: 

<img src='content_image/303502.jpg'>

and introduce one flavour of quark via a D7 brane probe in the geometry. The Dirac Born Infeld (DBI) action for the probe is 

<img src='content_image/303501.jpg'>

where P indicates the pullback of the spacetime metric onto the D7 world volume. The pullback of a metric is defined as 

<img src='content_image/303500.jpg'>

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 

with ω 2 = ρ 2 + | Φ | 2 = ρ 2 + ω 2 5 + ω 2 $_{6}$. 

<img src='content_image/303503.jpg'>

