Deformations in G_2 Manifolds

TL;DR

This paper investigates the deformation theory of associative submanifolds within G_2 manifolds, introducing a new approach that simplifies existing complex methods by using only the cross product operation and related algebraic structures.

Contribution

It provides new insights and results on G_2 manifold deformations using an elementary, self-contained approach centered on the cross product and algebraic identities.

Findings

Relates deformations to Seiberg-Witten type equations.

Simplifies proofs of known results like McLean's theorem.

Provides new deformation results using elementary methods.

Abstract

Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form \phi. A choice of 2-plane field \Lambda on M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relate the deformations to Seiberg-Witten type equations. Here all the surveyed results as well as the new ones about G_2 manifolds are proved by using only the cross product operation (equivalently \phi). We feel that mixing various different local identifications of the rich G_2 geometry (e.g. cross product, representation theory and the algebra of octonions) makes the study of G_2 manifolds looks harder then it is (e.g. the proof of McLean's theorem \cite{m}). We believe the approach here makes things easier and keeps the presentation elementary. This paper is essentially self contained.