Complex Structures, T-duality and Worldsheet Instantons in Born Sigma Models

We investigate doubled (generalized) complex structures in $2D$-dimensional Born geometries where T-duality symmetry is manifestly realized. We show that K\"{a}hler, hyperk\"{a}hler, bi-hermitian and bi-hypercomplex structures of spacetime are implemented in Born geometries as doubled structures. We find that the Born structures and the generalized K\"{a}hler (hyperk\"{a}hler) structures appear as subalgebras of bi-quaternions and split-tetra-quaternions. We find parts of these structures are classified by Clifford algebras. We then study the T-duality nature of the worldsheet instantons in Born sigma models. We show that the instantons in K\"{a}hler geometries are related to those in bi-hermitian geometries in a non-trivial way.