nstantons and Branes in Manifolds with Vector Cross Products

Abstract

In this paper we study the geometry of manifolds with vector cross product 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 G2manifolds.Second we classify Kähler manifolds with the complex analog of 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.Third we prove that, given any Calabi-Yau manifold, 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.

On topological $M$-theory

“…On Hermitian manifolds (g, J, M ), one can introduce the complex version [26] of cross vector products (see Appendix B for the definition). In this case, the complex vector product 3 is given by a holomorphic p-form which is either a holomorphic volume form or a holomorphic symplectic form on M .…”

“…The complex version of vector cross product has been introduced in [26]. Consider a Hermitian manifold (g, J, M ) and define the complex vector cross product as a holomorphic (p + 1)-form satisfying i e 1 ∧e 2 ∧···∧ep φ = 2 One can show from this definition that φ can be either a holomorphic symplectic form or a holomorphic volume form [26].…”

“…Consider a Hermitian manifold (g, J, M ) and define the complex vector cross product as a holomorphic (p + 1)-form satisfying i e 1 ∧e 2 ∧···∧ep φ = 2 One can show from this definition that φ can be either a holomorphic symplectic form or a holomorphic volume form [26]. Thus the examples of manifolds equipped with the complex vector cross product structure are hyperkähler and Calabi-Yau manifolds.…”

Thin instantons in $G_2$-manifolds and Seiberg-Witten invariants

“…complex surface) in X. In [25] J.H. Lee and the first author showed that the isotropic knot spaceK S 1 X of X admits a natural holomorphic symplectic structure.…”

“…Lagrangian submanifolds) are instantons or associative submanifolds (resp. coassociative submanifolds or branes) in M [25]. In [17], the Fredholm theory for instantons with coassociative boundary conditions has been set up.…”