Associative Submanifolds of a G 2 Manifold

We study deformations of associative submanifolds of a G 2 manifold. We show that deformation spaces can be perturbed to be smooth and finite dimensional , and they can be made compact by constraining them with an additional equation and reducing it to Seiberg-Witten theory. This allows us to associate in-variants of certain associative submanifolds. More generally we apply this process to certain associated Grassmann bundles in order to assign invariants to G 2 manifolds. 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 can not 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 asso-ciative submanifolds. It is generally expected that this problem should be remedied by deforming the ambient G 2 structure (i.e. by deforming ϕ), but unfortunately this 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 process. Ideally one would want to define auxiliary objects analogous to almost complex structures 'compatible with ϕ', and then deform the associative submani-folds by the help of these objects in a larger class of manifolds, just like deforming the holomorphic curves in the larger class of manifolds (pseudo-holomorphic curves). We will take approach to obtain the smoothness and the finite dimensionality of the moduli space. To get compactness of the moduli space we will relate the deformation equation to the Seiberg-Witten equations. Metric deformations also comes handy at the end in defining an invariant for G 2 manifolds since they help to increase the dimension of the relevant moduli space. In [AS] in order to carry this program we introduced complex associative submani-folds of G 2 manifolds, they are associative submanifolds whose normal bundles carry