Supergravity solutions for D$p$-D$(6-p)$ bound states: from $p=7$ to $p=-1$

Near horizon geometries of D$p$-branes with $p\neq 3$ are singular with a running dilaton. Bound states of D$p$ branes with their magnetic cousins, D$(6-p)$ branes, can stabilise the dilaton such that an AdS factor might appear in the near horizon region, potentially leading to a chain of AdS vacua of the form $AdS_{p+2}\times S^{p+2} \times \mathbb{T}^{6-2p}$. The solutions with $p=-1, 1, 3$ are supersymmetric with the cases $p=1, 3$ being well-known examples already. We construct the explicit (partially smeared) brane bound state solutions for all such configurations where the cases $p=-1,2$ are entirely novel and we find no AdS geometry for these. The two novel classes of solutions feature ghost branes (negative tension branes), and we suggest they are physical for the D$(-1)-$D$7$ solutions but unphysical for the D$2-$D$4$ solutions. The bound state of a D$(-1)$ and a D7 brane in supergravity was only hinted upon recently in \cite{Aguilar-Gutierrez:2022kvk} and here we correct the solution in order to preserve supersymmetry and find that the dilaton can indeed be stabilised which indicates there could be a holographic dual matrix theory which generalises the IKKT matrix model to allow for conformal invariance.