Associative Submanifolds of a G 2 Manifold

We study deformations of associative submanifolds Y 3 ⊂ 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. McLean showed that, in a G 2 manifold (M 7 , ϕ), the space of associative submani-folds near a given one Y 3 , can be identified with the harmonic spinors on Y twisted by a certain bundle E (the kernel of a twisted Dirac operator) [M]. But since we cannot control the cokernel of the Dirac operator (it has index zero), the dimension of its kernel might vary. This is the obstruction to smoothness of the moduli space of associative submanifolds. This problem can be remedied either by deforming the ambient G 2 structure (i.e. by deforming ϕ) or by deforming the connection in the normal bundle [AS]. The first process might move ϕ to a non-integrable G 2 structure. If we are to view (M, ϕ) as an analogue of a symplectic manifold and ϕ a symplectic form, and view the associative submanifolds as analogues of holomorphic curves, deforming ϕ would be too destructive. In the second process we use the connections as auxiliary objects to deform the associative submanifolds in a larger space, just like deforming the holomorphic curves by using almost complex structures (pseudo-holomorphic curves). By this approach we obtain the smoothness of the moduli space. We get compactness by relating the deformation equation to the Seiberg-Witten equations. In this paper we summarize the results of [AS] where we introduced complex asso-ciative submanifolds of G 2 manifolds; they are associative submanifolds whose normal bundles carry a U(2) structure. This is no restriction, since every associative subman-ifold has this structure, but if we require that their deformations be compatible with the background connection we must have an integrability condition, i.e. the condition that the connection on their normal bundles (induced by the G 2 background metric)