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Hyperkähler, Bi-hypercomplex, Generalized Hyperkähler Structures and T-duality

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