Harvey Lawson Manifolds and Dualities

The purpose of this paper is to introduce Harvey-Lawson manifolds and review the construction of certain mirror dual Calabi-Yau submanifolds inside a G_2 manifold. More specifically, given a Harvey-Lawson manifold HL, we explain how to assign a pair of tangent bundle valued 2 and 3-forms to a G_2 manifold (M,HL, \varphi, \Lambda), with the calibration 3-form \varphi and an oriented 2-plane field \Lambda. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional sub bundles of T(M). When these bundles are integrated they give mirror CY manifolds (related thru HL manifolds).