Finding $AdS^{5} \times S^{5}$ in 2+1 dimensional SCFT physics

We study solutions of type IIB string theory dual to ${\cal N}=4$ supersymmetric Yang-Mills theory on half of $\mathbb{R}^{3,1}$ coupled to holographic three-dimensional superconformal field theories (SCFTs) at the edge of this half-space. The dual geometries are asymptotically $AdS^5 \times S^5$ with boundary geometry $\mathbb{R}^{2,1} \times \mathbb{R}^+$, with a geometrical end-of-the-world (ETW) brane cutting off the other half of the asymptotic region of the would-be Poincare$AdS^5 \times S^5$. We show that by choosing the 3D SCFT appropriately, this ETW brane can be pushed arbitrarily far towards the asymptotic region, recovering the missing half of Poincare$AdS^5 \times S^5$. We also show that there are 3D SCFTs whose dual includes a wedge of Poincare$AdS^5 \times S^5$ with an angle arbitrarily close to $\pi$, with geometrical ETW branes on either side.