Higher dimensional knot spaces for manifolds with vector cross product

Vector cross product structures on manifolds include symplectic, volume, G2and Spin (7)-structures. We show that their knot spaces have natural symplectic structures, and we relate instantons and branes in these manifolds with holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi-Yau manifold de…nes a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi-Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.