Deformations in G2 manifolds

Here we study the deformations of associative submanifolds inside a G2 manifold M7 with a calibration 3-form φ. A choice of 2-plane field Λ on M (which always exists) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM=E⊕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 G2 manifolds are proved by using only the cross product operation (equivalently φ). We feel that mixing various different local identifications of the rich G2 geometry (e.g. cross product, representation theory and the algebra of octonions) makes the study of G2 manifolds look harder then it is (e.g. the proof of McLean's theorem [R.C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998) 705–747]). We believe the approach here makes things easier and keeps the presentation elementary.