A ug 2 00 7 CALIBRATED MANIFOLDS AND GAUGE THEORY

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G2-manifold (M, ϕ) can be identified with the kernel of a Dirac operator D / : Ω 0 (ν) → Ω 0 (ν) on the normal bundle ν of Y. Here, we generalize this to the 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. Infinitesi-mally, this corresponds to twisting the Dirac operator D / → D / A with connections A of ν. Furthermore, the normal bundles of the associative submanifolds with Spin c structure have natural complex structures, which helps us to relate their deformations to Seiberg-Witten type equations. 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 (almost) Λ-associative submanifolds of M , whose deformation equations, when perturbed, reduce to Seiberg-Witten equations, hence we can assign local invariants to these subman-ifolds. Using this we can assign an invariant to (M, ϕ, Λ). These Seiberg-Witten equations on the submanifolds are restrictions of global equations on M. We also discuss similar results for the Cayley submanifolds of a Spin(7) manifold. 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 by 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 T M). Infinitesimally, 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).