Instantons and branes in manifolds with vector cross product

TL;DR

This paper explores the geometry of manifolds with vector cross product, developing theories of instantons and branes, classifying complex analogs like Calabi-Yau and hyperkähler manifolds, and linking Calabi-Yau geometry to holomorphic symplectic structures on knot spaces.

Contribution

It introduces a comprehensive framework for instantons and branes in manifolds with vector cross product and classifies complex analogs such as Calabi-Yau and hyperkähler manifolds.

Findings

Classification of Kahler manifolds with complex vector cross product as Calabi-Yau and hyperkähler.

Identification of instantons and branes as special submanifolds like Lagrangian and associative submanifolds.

Existence of a natural holomorphic symplectic structure on isotropic knot spaces of Calabi-Yau manifolds.

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 G_2-manifolds. Second we classify Kahler manifolds with the complex analog of vector cross product, namely they are Calabi-Yau manifolds and hyperkahler 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 pi/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.