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

TL;DR

This work constructs and analyzes supersymmetric orbifolds built from weighted projective spaces ${\mathbb{WCP}^2}$ and ${\mathbb{WCP}^3}$, realized as U(1) orbifold bundles over odd spheres and explored via two parametrisations of ${\mathbb{WCP}^3}$. The authors establish precise conditions under which these orbifolds preserve portions of supersymmetry, including when KK reductions to round ${\mathbb{WCP}^n}$ are possible, and they show that for certain weight tunings SUSY persists beyond gauged supergravity. They then generate a variety of AdS near-horizon orbifolds of canonical brane geometries (D3, D1-D5, M2, M5) and, using string dualities, obtain AdS solutions with round ${\mathbb{WCP}^n}$ factors such as AdS$_7\times{\mathbb{WCP}^1}$, AdS$_5\times{\mathbb{WCP}^2}$, and AdS$_4\times{\mathbb{WCP}^3}$, with SUSY determined by weight choices. The paper also constructs an AdS$_3$ solution featuring a topological ${\mathbb{WT}^{(1,1)}}$ orbifold, including IIA realizations and central charge computations, highlighting how orbifold data control holographic characteristics. Overall, the results provide a versatile framework for AdS/CFT constructions incorporating ${\mathbb{WCP}^n}$ factors and point to rich CFT duals and potential extensions to broader orbifold families.

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, round $\mathbb{WCP}^{2}$ 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. We also leverage our results to construct a supersymmetric AdS$_3$ solution containing a topological $\mathbb{T}^{(1,1)}$ space with 4 orbifold singularities.