  By a theorem of Mclean, the deformation space of an associative submanifold Y
of an integrable G_2 manifold (M,\phi) can be identified with the kernel of a
Dirac operator D:\Omega^{0}(\nu) -->\Omega^{0}(\nu) on the normal bundle \nu 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 D --> D_{A}
with connections A of \nu. 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,\phi, \Lambda) (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) \Lambda-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,\phi, \Lambda). 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.
