Second order deformations of associative submanifolds in nearly parallel $G_2$-manifolds

TL;DR

This paper investigates second order deformations of associative submanifolds in nearly parallel G2-manifolds, providing explicit conditions for their integrability and demonstrating unobstructed deformations for a specific homogeneous example.

Contribution

It offers a necessary and sufficient condition for second order integrability of associative deformations, extending previous infinitesimal deformation analysis.

Findings

Derived explicit second order integrability conditions.

Proved unobstructed second order deformations for a specific homogeneous submanifold.

Extended deformation theory in nearly parallel G2-geometry.

Abstract

Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in ${\rm Spin(7)}$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associative submanifolds were considered by the author. This paper is a continuation of the work. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly. As an application, we show that the infinitesimal deformations of a homogeneous associative submanifold in the 7-sphere given by Lotay, which he called $A_3$, are unobstructed to second order.