Hyperkähler, bi-hypercomplex, generalized hyperkähler structures and T-duality

We exploit the doubled formalism to study comprehensive relations among T-duality, complex and bi-hermitian structures (J+,J−) in two-dimensional N=(2,2) sigma models with/without twisted chiral multiplets. The embedded generalized Kähler are organized into algebra of tri-complex numbers. write down an analogue Buscher rule by which T-duality transformation apparent. also bi-hypercomplex hyperkähler cases N=(4,4) theories. They expressed, as a covariant fashion, form split-bi-quaternion algebras. As concrete example, we show explicit relation between KK-monopole (Taub-NUT space) H-monopole (smeared NS5-brane). Utilizing this result, comment on for worldsheet instantons these geometries.