Are there any flow to different dimension? So far as I know from gauged supergravity there are studies of RG flows from the same dimension. Are there any flow to different dimension, i.e. flow from $AdS_5$ to $AdS_4$?
 A: As far as I know, in gauged supergravity, there are domain wall solutions which are dual to renormalization group flow in the field theory side of the AdS/CFT correspondance. These domain wall solutions connect different vacua of the same gauged SUGRA theory (for example, those vacua with different number of supersymmetries preserved like $AdS_5$ with $\mathcal N = 8$ to another $AdS_5$ solution with $\mathcal N=2$), so in that respect, there are domain walls connecting different $AdS$ vacua in the same 5D. There are also domain walls connecting $AdS_{5-p}\times \Sigma^p$ for $p=2,3$ at the near-horizon limit to the full $AdS_5$ spacetime at infinity. These are known as RG flow across dimensions on the field theory side. However, I have not seen domain wall connecting $AdS_5$ to $AdS_4$ at all. The reason being that to support the internal space $\Sigma_p$, some gauge fields need to be turned on, and the "smallest" gauge field is a $U(1)$ gauge field, which can support an $S^2$ factor. 
