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

TL;DR

This paper constructs supersymmetric orbifolds of higher-dimensional weighted projective spaces, exploring their properties and applications in supergravity and AdS/CFT, including new solutions with orbifold singularities.

Contribution

It introduces supersymmetric constructions of weighted projective spaces $ ext{WCP}^n$, extending supersymmetry beyond gauged supergravity and enabling new supergravity solutions.

Findings

Round $ ext{WCP}^2$ and $ ext{WCP}^3$ are compatible with supersymmetry beyond gauged supergravity.

Construction of supersymmetric solutions like AdS$_5 imes ext{WCP}^2 imes S^1$ and AdS$_4 imes ext{WCP}^3$.

Development of a supersymmetric AdS$_3$ solution with orbifold singularities.

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.