M ay 2 00 6 CALIBRATED MANIFOLDS AND GAUGE THEORY

By a theorem of Mclean, the deformation space of an associative sub-manifolds of an integrable G2 manifold (M, ϕ) at Y ⊂ M can be identified with the kernel of the Dirac operator D / : Ω 0 (ν) → Ω 0 (ν) on the normal bundle ν of Y. We generalize this to non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through 'pseudo-associative' submanifolds. Infinitesimally this corresponds to twisting the Dirac operator D / → D / A by connections A of ν. If we consider G2 manifolds with 2-plane fields (M, ϕ, Λ) (they always exist) we can split the tangent space T(M) as a direct sum of an associative 3-plane bundle and a complex 4-plane bundle. This allows us to define 'complex associa-tive submanifolds' of M , whose deformation equations reduce to Seiberg-Witten equations, hence we can assign local invariants to these submanifolds. Using this we associate invariants to (M, ϕ, Λ). These Seiberg-Witten equations on the submanifolds are restrictions of global equations on M. We also discuss similar theorems for Cayley submanifolds of a Spin(7) manifold. 0. INTRODUCTION We first study deformations of associative submanifolds Y 3 of a G 2 manifold (M 7 , ϕ), where ϕ ∈ Ω 3 (M) is the G 2 structure. We prove a generalized version of the McLean's theorem where integrability condition of the underlying G 2 structure is not necessary. This deformation space might be singular, but we can perturbing it with some natural parameters it can be made smooth. This amounts to deforming Y through the associatives in (M, ϕ) with varying ϕ, or alternatively deforming Y through the pseudo-associative submanifolds (Y 's whose tangent planes become associative after rotating by a generic element of the gauge group of M). Infinites-imally these perturbed deformations correspond to the kernel of the twisted Dirac operator D / A : Ω 0 (ν) → Ω 0 (ν), twisted by some connection A in ν(Y). We can view (M, ϕ) as an analog of a symplectic manifold, and view a non-vanishing 2-plane field Λ on M as an analogue of a complex structure taming ϕ. Note that 2-plane fields are stronger versions of Spin c structures on M 7 , and they