Instantons and branes in manifolds with vector cross products

In this paper we study the geometry of manifolds with vector cross products and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and coassociative submanifolds in G₂-manifolds. Second we classify Kähler manifolds with the complex analog of the vector cross product, namely they are Calabi-Yau manifolds and hyperkähler manifolds. Furthermore we study instantons, Neumann branes and Dirichlet branes on these manifolds. For example they are special Lagrangian submanifolds with phase angle zero, complex hypersurfaces and special Lagrangian submanifolds with phase angle π/2 in Calabi-Yau manifolds.