the Constable-Myers solution, presented in chapter 4. 6.1.1 Truncating the Constable-Myers Geometry The truncation we propose is incredibly simple. For convenience we repro- duce the equations of motion (4.26), (4.29) for the eight dimensional fields defined on the surface of the D7 brane. The functions f$_{n}$ represents the pion fields, and g$_{n}$ the vector fields. G is defined as in (4.23). The reader is reminded that the metric is also defined in eight dimensions: Since we want to propose a theory that is dual to QCD, the fields must not have any components on the three sphere. The simplest way to do this is to claim that all of the fields f$_{n}$, g$_{n}$, φ and σ are constants on the 3-sphere. They are only functions of r and x only. 89 ds 2 8 D = $_{H}$− 1 / 2 ( ω 4 + b 4 ω 4 − b 4 ) δ/ 4 η$_{µν}$dx $^{µ}$dx$^{ν}$+ R $^{2}$H1 / 2 ( ω 4 + b 4 ω 4 − b 4 ) (2 − δ ) / 4 ω 4 − b 4 ω 4 4 ∑ i =1 dω 2 i (6.4) e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) $^{2}$M 2 $_{n}$R $^{2}$g$_{n}$ ( ρ ) H ( ω 4 − 1 ω 4 + 1 ) 1 4 + ∂$_{ρ}$ ( e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) $_{2}$∂$_{ρ}$ g$_{n}$ ( ρ ) ω 4 √ ( ω 4 − 1)( ω 4 + 1) ) = 0 (6.3) R $^{2}$M2 $_{n}$e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) $_{2}$H ( ( ρ 2 + σ 2 $_{0}$) 2 + 1 ( ρ 2 + σ 2 $_{0}$) 2 − 1 ) (1 − δ ) / 2 ( ρ 2 + σ 2 $_{0}$) 2 − 1 ( ρ 2 + σ 2 $_{0}$) 2 σ 2 $_{0}$f$_{n}$ + ∂$_{ρ}$ ( e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ 2 $_{0}$) $_{2}$σ 2 $_{0}$∂$_{ρ}$ f$_{n}$ ) = 0 (6.2)