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CALIBRATED MANIFOLDS AND GAUGE THEORY SELMAN AKBULUT AND SEMA SALUR
 

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 G2-manifold (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 non-integrable case, and also show that the defor-
mation 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 . Furthermore, the normal bundles of the associative submanifolds with
Spinc
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, , ) (they always exist)
we can split the tangent space TM as a direct sum of an associative 3-plane bun-
dle and a complex 4-plane bundle. This allows us to define (almost) -associative
submanifolds of M, whose deformation equations, when perturbed, reduce to
Seiberg-Witten equations, hence we can assign local invariants to these subman-

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics