m at h . D G ] 1 4 Ju n 20 07 MIRROR DUALITY VIA G 2 AND Spin ( 7 ) MANIFOLDS

The main purpose of this paper is to give a construction of certain " mirror dual " Calabi-Yau submanifolds inside of a G2 manifold. More specifically, we explain how to assign a G2 manifold (M, ϕ, Λ), with the calibration 3-form ϕ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional 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 (N 8 , Ψ). In case N 8 admits an oriented 3-plane field, by iterating this process we obtain Calabi-Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G2 (or Spin(7)) manifold.