Calibrated Manifolds and Gauge Theory

Abstract

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,ϕ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof 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. Infinitesimally, 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 G_2 manifolds with 2-plane fields (M,ϕ, Λ) (they always exist) we can split the tangent space TM 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 submanifolds. 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.