AdS$_3$ vacua realising $\mathfrak{osp}(n|2)$ superconformal symmetry

We consider ${\cal N}=(n,0)$ supersymmetric AdS$_3$ vacua of type II supergravity realising the superconformal algebra $\mathfrak{osp}(n|2)$ for $n>4$. For the cases $n=6$ and $n=5$, one can realise these algebras on backgrounds that decompose as foliations of AdS$_3\times \mathbb{CP}^3$ ( squashed $\mathbb{CP}^3$ for $n=5$) over an interval. We classify such solutions with bi-spinor techniques and find the local form of each of them: They only exist in (massive) IIA and are defined locally in terms of an order 3 polynomial $h$ similar to the AdS$_7$ vacua of (massive) IIA. Many distinct local solutions exist for different tunings of $h$ that give rise to bounded (or semi infinite) intervals bounded by physical behaviour. We show that it is possible to glue these local solutions together by placing D8 branes on the interior of the interval without breaking supersymmetry, which expands the possibilities for global solutions immensely. We illustrate this point with some simple examples. Finally we also show that AdS$_3$ vacua for $n=7,8$ only exist in $d=11$ supergravity and are all locally AdS$_4\times$S$^7$.