Calibrated manifolds and Gauge theory

Abstract

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 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. Infinitesimally, this corresponds to twisting the Dirac operator 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 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.