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The choice of m$_{5}$ for each field is determined by the relation (Δ − p )(Δ+ p − 4) = m 2 5 which we discussed in section 2.7. The metric, also an ans¨ atz, is simply AdS$_{5}$ , which is the simplest choice we could make. In keeping with [2], in this section we will write the fifth dimension in terms of z , instead of the usual r . The two are related by z = r − $^{1}$. Hence z ∼ 0 is the ultraviolet part of the spacetime, and z ∼ z$_{m}$ is the infrared of the spacetime. 

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The use of an AdS geometry implies that the dual theory will be conformal - not something we want in a dual to QCD. So to give the theory a mass gap, the spacetime is defined to end at some IR scale z = z$_{m}$ . In the window 0 < z ⩽ z$_{m}$ , the background gauge is conformal, and the coupling doesn’t run. 

The 5D action is 

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where D$_{µ}$X = ∂$_{µ}$X − iA$_{Lµ}$X + iXA$_{Rµ}$ , A$_{L,R}$ = A a $_{L,R}$t$^{a}$, and F$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ − i [ A$_{µ}$, A$_{ν}$ ]. X is a Higgs-like field, holographically dual to the quark mass and condensate. Its value in the UV will be: 

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