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 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 The DBI action for (3.5) is 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. where the ten-dimensional metric has been written as 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 49 P [ G ] ≡ ∂x α ∂ξ a ∂x β ∂ξ b G$_{ab}$ (3.4) S$_{D}$$_{7}$ = − T$_{7}$ ∫ d $^{8}$ξ √ − det ( P [ G ]) (3.5) − T$_{7}$ ∫ d $^{8}$ξ ρ 3 R 3 √ 1 + ( ∂u$_{5}$ ∂ρ 2 + ∂u$_{6}$ ∂ρ 2 ) + R 2 u 2 ( ∂u$_{5}$ ∂x 2 + ∂u$_{6}$ ∂x 2 ) (3.6) ds 2 = u 2 R $_{2}$η$_{µν}$ dx $^{µ}$dxν + R 2 u 2 ( dρ 2 + ρ $^{2}$d Ω 2 3 + du 2 5 + du 2 6 ) (3.7)