A Construction of Killing Spinors on S^n

TL;DR

The paper provides explicit, all-n dimensions expressions for Killing spinors on the sphere $S^n$ and extends these results to Killing spinors on AdS×S backgrounds and hyperbolic spaces $H^n$. By expressing the $S^n$ spinors as a product of exponentials in gamma-matrix space, the authors derive a concise, universal construction and prove its validity with a careful spin-connection analysis. They then combine these sphere spinors with AdS Killing spinors to produce explicit forms for AdS$_m\times S^n$ backgrounds appearing in supergravity, including AdS$_5\times S^5$, AdS$_4\times S^7$, and AdS$_7\times S^4$, among others. The work enables direct generation of Killing vectors from spinor bilinears and has practical impact for supersymmetry analyses and AdS/CFT applications. Overall, it supplies a powerful, explicit toolkit for Killing spinors across maximally symmetric spaces and their AdS extensions.

Abstract

We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdS x Sphere supergravity backgrounds, including AdS_5 x S^5$, AdS_4 x S^7 and AdS_7 x S^4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces H^n.