Associative submanifolds of a G2 manifold

TL;DR

This paper investigates the deformation theory of associative submanifolds within G2 manifolds, demonstrating how to smooth, compactify, and assign local invariants to these submanifolds through global equations.

Contribution

It introduces a method to perturb deformation spaces of associative submanifolds to be smooth, compact, and zero-dimensional, enabling the definition of local invariants.

Findings

Deformation spaces can be made smooth and compact.

Local invariants can be associated to associative submanifolds.

Global equations on Grassmann bundles govern local deformation conditions.

Abstract

We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of $M$. The local equations at each associative $Y$ are restrictions of a global equation on a certain associated Grassmann bundle over $ M$.