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A not very rigorous answer to your question is that yes, the metric encodes information needed to determine if the theory is in a confined phase or not. If the spacetime is asymptotically locally AdS, or perhaps a variant thereof that would still be expected to be dual to a field theory (a good example is Klebanov-Strassler), then the test is to see if the ...

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The "singleton" and "doubleton" language comes from the oscillator method of finding group representations. Given a (non-compact) group $G$ admitting lowest weight representation with maximal compact subgroup of the form $H\times\mathrm{U}(1)$ for some other compact group $H$, the oscillator method is to describe (unitary irreducible) representations by ...

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Let $\Omega^2=\exp\phi/2$ so that $\gamma=\Omega^2 g$. Then we have the standard formula $$R_\gamma=\Omega^{-2}\big(R_g-2(n-1)\Delta\ln\Omega-(n-2)(n-1)[\nabla\ln\Omega]^2\big)$$ This is proven in any number of General Relativity texts, but the one in R.M. Wald, General Relativity (1984), is particularly easy to follow and is proved in the form shown here. ...

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