 
Summary: CALIBRATED MANIFOLDS AND GAUGE THEORY
SELMAN AKBULUT AND SEMA SALUR
Abstract. By a theorem of Mclean, the deformation space of an associative
submanifold Y of an integrable G2manifold (M, ) can be identified with the
kernel of a Dirac operator D/ : 0
() 0
() on the normal bundle of Y .
Here, we generalize this to the nonintegrable case, and also show that the defor
mation space becomes smooth after perturbing it by natural parameters, which
corresponds to moving Y through `pseudoassociative' submanifolds. Infinitesi
mally, this corresponds to twisting the Dirac operator D/ D/ A with connections
A of . Furthermore, the normal bundles of the associative submanifolds with
Spinc
structure have natural complex structures, which helps us to relate their
deformations to SeibergWitten type equations.
If we consider G2 manifolds with 2plane fields (M, , ) (they always exist)
we can split the tangent space TM as a direct sum of an associative 3plane bun
dle and a complex 4plane bundle. This allows us to define (almost) associative
submanifolds of M, whose deformation equations, when perturbed, reduce to
SeibergWitten equations, hence we can assign local invariants to these subman
