2 M ay 2 00 5 CALIBRATED MANIFOLDS AND GAUGE THEORY

We show that the moduli spaces of associative submanifolds of a G 2 manifold (and Cayley submanifolds of a Spin(7) manifold) can be perturbed to smooth manifolds. By using connections as natural parameters and by constraining them with an additional equation and using Seiberg-Witten theory we can make them compact, and hence assign local invariants to these submanifolds. The local equations of calibrated submanifolds are restrictions of a global equation on a certain associated Grassmann bundle over the ambient G 2 (or Spin(7)) manifold. 0. INTRODUCTION Here we first study deformations of associative submanifolds of a G 2 manifold (M 7 , ϕ), where ϕ ∈ Ω 3 (M) is the 3-form defining the G 2 structure on M. Here we view (M, ϕ) as an analog of a symplectic manifold and ϕ a symplectic form, and view the associative submanifolds as analogs of holomorphic curves. We define auxiliary objects (universal connections) analogous to almost complex structures compatible with ϕ, and then perturb the space of associative submanifolds Y 3 ⊂ M 7 to a smooth manifold with the help of these objects. This is anologous to deforming the holomorphic curves to pseudo-holomorphic curves. By this approach we obtain the smoothness and the finite dimensionality of the perturbed moduli space. By relating the deformation equation to Seiberg-Witten equations we get the local compactness, hence local invariants. There is a similar story for the deformations of Cayley sub-manifolds X 4 ⊂ N 8 of a Spin(7) manifold (N 8 , Ψ), which we discuss at the end. So in a way associative and Cayley manifolds in G 2 and Spin(7) manifolds behave much like higher dimensional analogues of holomorphic curves in a Calabi Yau manifolds. To carry out this process we introduce the notion of complex associative (or complex Cayley) submanifolds of a G 2 (or a Spin(7)) manifold; they are associative and Cayley submanifolds whose normal bundles carry a U(2) structure. This structure can be induced from a global U(2) structure on the ambient G 2 (or Spin(7)) manifold. This is not much topological restriction on the submanifold, but if we require that their