Deformations in G2 Manifolds

Here we study the deformations of associative submanifolds inside a G2 manifold M 7 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 SeibergWitten 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 looks harder then it is (e.g. the proof of McLean’s theorem [M]). We believe the approach here makes things easier and keeps the presentation elementary. This paper is essentially self-contained. 1. G2 manifolds We first review the basic results about G2 manifolds, along the way we give a self-contained proof of the McLean’s theorem and its generalization [M], [AS1]. A G2 manifold (M,φ,Λ) with an oriented 2-plane field gives various complex structures on some of the subbundles of TM . This imposes interesting structures on the deformation theory of its associative submanifolds. By using this we relate them to the Seiberg-Witten type equations. Let us recall some basic definitions (c.f. [B1], [B2],[HL]): Octonions give an 8 dimensional division algebra O = H⊕lH = R generated by 〈1, i, j, k, l, li, lj, lk〉. The imaginary octonions imO = R is equipped with the cross product operation × : R×R → R defined by u× v = im(v̄.u). The exceptional Lie group G2 is the linear automorphisms of imO preserving this cross product. It can also be defined in terms of the orthogonal 3-frames: (1) G2 = {(u1, u2, u3) ∈ (imO) 3 | 〈ui, uj〉 = δij , 〈u1 × u2, u3〉 = 0 }. Date: September 16, 2007.