$\mathcal{H}$-Killing Spinors and Spinorial Duality for Homogeneous 3-$(\alpha ,\delta )$-Sasaki Manifolds

We show that $3$-$(\alpha ,\delta )$-Sasaki manifolds admit solutions of a certain new spinorial field equation (the $\mathcal{H}$-Killing equation) generalizing the well-known Killing spinors on $3$-Sasakian manifolds. These $\mathcal{H}$-Killing spinors have more desirable geometric properties than the spinors obtained by simply deforming a $3$-Sasakian metric; in particular we obtain a one-to-one correspondence between $\mathcal{H}$-Killing spinors on dual pairs of homogeneous $3$-$(\alpha ,\delta )$-Sasaki spaces. Finally, we show that $\mathcal{H}$-Killing spinors generalize certain special spinors in dimension $7$ previously constructed by Agricola-Friedrich and Agricola-Dileo using ${\text{G}}_2$-geometry.