arXiv:math/0605277v1 [math.DG] 10 May 2006 MIRROR DUALITY AND G 2 MANIFOLDS SELMAN AKBULUT AND SEMA SALUR Abstract. Let (M,ϕ) be a G2 manifold with the calibration 3-form ϕ. The purpose of this note is to explain briefly how to associate ϕ parametrized tan- gent bundle valued 2 and 3-forms on M, and and their relations to the complex and symplectic structures on the certain underlying 6-dimensional bundles of T (M). These 6-dimensional subbundles gives different complex and symplectic structures. For example, in the special case of M =Calabi-Yau×S 1 , one of the 6-dimensional subbundle corresponds to the tangent bundle of the CY manifold. This explains the mirror duality between the symplectic and complex structures on the CY 3-folds inside of a G2 manifold. One can easilyextend these arguments to noncompact G2 manifolds of the form CY×R. 1. Introduction Let (M 7 ,ϕ) be a G 2 manifold with the calibration 3-form ϕ. If ϕ restricts to be the volume form of a 3-dimensional submanifold Y 3 , then Y is called an associative submanifold of M . Associative submanifolds are very interesting objects as they act very similar to the holomorphic curves of Calabi-Yau manifolds. Two years ago in [AS], we began a program in order to understand the deforma- tions of associative submanifolds. Our main goal was to construct Gromov-Witten like invariants for G 2 manifolds from its associative submanifolds. One of our main observations was that oriented 2-plane fields on M always exist by a theorem of Thomas [T] and by using them one can split the tangent bundle T (M )= E ⊕ V as a direct sum of an associative 3-plane bundle E and a complex 4-plane bundle V. This allowed us to define ‘complex associative submanifolds’ of M , whose defor- mation equations may be reduced to the Seiberg-Witten equations, and hence we can assign local invariants to them, and assign various invariants to (M,ϕ, Λ). It turns out that these Seiberg-Witten equations on the submanifolds are restrictions of global equations on M . Recently, we realized that the rich geometric structures of G 2 manifolds (M,ϕ) provide complex and symplectic structures to certain 6-dimensional subbundles of T (M ). In this brief note we explain how these structures are related to the mirror Date : September 16, 2018. 1991 Mathematics Subject Classification. 53C38, 53C29, 57R57. Key words and phrases. mirror duality, calibration. First named author is partially supported by NSF grant DMS 9971440. 1