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

Abstract

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.

arXiv comment23 pages, no figure, a reference added, minor modifications, version appeared in Nucl. Phys. B

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Cited by in corpus (3)

• On Geometries and Monodromies for Branes of Codimension Two
• T-duality on Almost Hermitian Spaces
• Doubled space and extended supersymmetry