Supersymmetric ${\mathbb{WCP}}^n$, AdS near horizons and orbifolds

Abstract

We construct and study the supersymmetry properties of the weighted projective spaces ${\mathbb{WCP}}^2$ and ${\mathbb{WCP}}^3$. These are topologically ${\mathbb{CP}}^n$ with $n+1$ orbifold singularities and as such are higher dimensional analogues of the ``spindle'' or ${\mathbb{WCP}}^1$. We use these to construct interesting supersymmetric orbifolds of canonical near horizon geometries of relevance to the AdS/CFT correspondence. Interestingly, for certain tunings of their integer weights, and unlike the spindle, ${\mathbb{WCP}}^2$ and ${\mathbb{WCP}}^3$ are compatible with supersymmetry beyond the realm of gauged supergravity. This allows one to construct interesting supersymmetric solutions in type II supergravity such as AdS${}_5\times {\mathbb{WCP}}^2\times {\text{S}}^1$ and AdS${}_4\times {\mathbb{WCP}}^3$ via duality, in which ${\mathbb{WCP}}^n$ appears without a connection fibred over it. We also leverage our results to construct a supersymmetric AdS${}_3$ solution containing a topological ${\mathbb{T}}^{(1,1)}$ space with 4 orbifold singularities.

v3: Published version