Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds

TL;DR

This paper studies how Einstein deformations affect Killing spinors on Sasakian and 3-Sasakian manifolds, revealing that the number of preserved spinors can decrease under deformation.

Contribution

It identifies specific infinitesimal Einstein deformations that alter the number of Killing spinors, including explicit examples on 3-Sasakian manifolds.

Findings

Deformations can preserve, reduce, or eliminate Killing spinors.

The number of Killing spinors is upper semi-continuous under Einstein deformations.

Toric 3-Sasakian manifolds provide examples with exactly two preserved Killing spinors.

Abstract

We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.