Infinitesimal deformations of Killing spinors on nearly parallel $\mathrm{G}_2$-manifolds

TL;DR

This paper investigates infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds, linking these deformations to Einstein metric variations and analyzing associated eigenspaces of the Laplacian.

Contribution

It provides a detailed analysis of how infinitesimal Killing spinor deformations correspond to nearly parallel G2-structure variations and identifies the Rarita-Schwinger fields within Laplacian eigenspaces.

Findings

Deformations of Killing spinors correspond to deformations of G2-structures.

The space of Rarita-Schwinger fields matches a Laplacian eigenspace.

Infinitesimal deformations relate to Einstein metric variations.

Abstract

Manifolds admitting Killing spinors are Einstein manifolds. Thus, a deformation of a Killing spinor entails a deformation of Einstein metrics. In this paper, we study infinitesimal deformations of Killing spinors on nearly parallel $\mathrm{G}_2$-manifolds. Since there is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$-structures and Killing spinors on 7-dimensional spin manifolds, our results imply that infinitesimal deformations of nearly parallel $\mathrm{G}_2$-structures are examined in terms of Killing spinors. Applying the same technique, we identify that the space of the Rarita-Schwinger fields coincides with a subspace of the eigenspace of the Laplacian.