content_image/308257.jpg content_image/308258.jpg content_image/308259.jpg content_image/308260.jpg
gauge field U(1) living on the D7 brane. P [ G ] is the pullback of the ten dimensional metric onto the surface of the D7 brane. It is defined as 

<img src='content_image/308257.jpg'>

with α, β running over the coordinates on the D7 brane, and a, b running over the whole spacetime’s coordinates. 

For now, we neglect the U(1) field and the Wess-Zumino term, leaving us with 

<img src='content_image/308258.jpg'>

We can neglect the Wess-Zumino term because it is only relevant for gauge fields with a vector index on the 5-sphere. In other words they carry a supersymmetry charge. Such states are important in the full dual, but for this thesis they are unimportant - we are trying to provide holographic descriptions of QCD, which does not (to the best of our knowledge) contain any supersymmetric states. We will re-introduce the U(1) field in section 3.2.2. It will turn out to be holographic to the vector meson sector. 

The DBI action for (3.5) is 

<img src='content_image/308259.jpg'>

where the ten-dimensional metric has been written as 

<img src='content_image/308260.jpg'>

with u 2 = ρ 2 + u 2 5 + u 2 $_{6}$. The x direction is perpendicular to the D7’s world volume, so fluctuations in x correspond to the meson spectrum. We come to this next, but first we address fluctuations in the ρ direction - this governs 

