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

TL;DR

This paper introduces a new class of spinorial solutions called $\\mathcal{H}$-Killing spinors on 3-$(\alpha,\delta)$-Sasaki manifolds, revealing their geometric properties, duality relations, and connections to G2-geometry.

Contribution

It defines and studies $\\mathcal{H}$-Killing spinors, establishing their properties, duality correspondence on homogeneous spaces, and relation to previously known G2-structure spinors.

Findings

Existence of $\\mathcal{H}$-Killing spinors on 3-$(\alpha,\delta)$-Sasaki manifolds.

One-to-one correspondence between $\\mathcal{H}$-Killing spinors on dual homogeneous spaces.

Connection of $\\mathcal{H}$-Killing spinors to special G2-structure spinors in dimension 7.

Abstract

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.