 
Summary: MIRROR DUALITY VIA G2 AND Spin(7) MANIFOLDS
SELMAN AKBULUT AND SEMA SALUR
Abstract. The main purpose of this paper is to give a construction of certain
"mirror dual" CalabiYau submanifolds inside of a G2 manifold. More specifically,
we explain how to assign a G2 manifold (M, , ), with the calibration 3form
and an oriented 2plane field , a pair of parametrized tangent bundle valued 2
and 3forms of M. These forms can then be used to define different complex and
symplectic structures on certain 6dimensional subbundles of T(M). When these
bundles are integrated they give mirror CY manifolds. In a similar way, one can
define mirror dual G2 manifolds inside of a Spin(7) manifold (N8
, ). In case N8
admits an oriented 3plane field, by iterating this process we obtain CalabiYau
submanifold pairs in N whose complex and symplectic structures determine each
other via the calibration form of the ambient G2 (or Spin(7)) manifold.
1. Introduction
Let (M7, ) be a G2 manifold with the calibration 3form . If restricts to be
the volume form of an oriented 3dimensional submanifold Y 3, then Y is called an
associative submanifold of M. Associative submanifolds are very interesting objects
as they behave very similarly to holomorphic curves of CalabiYau manifolds.
In [AS], we studied the deformations of associative submanifolds of (M, ) in order
