Hyperk\"{a}hler, Bi-hypercomplex, Generalized Hyperk\"{a}hler Structures and T-duality

TL;DR

This paper explores the relations between T-duality and complex structures in supersymmetric sigma models, introducing a new T-duality rule and analyzing hyperk"ahler structures, with explicit examples like the KK-monopole and H-monopole.

Contribution

It introduces an analogue of the Buscher rule for bi-hermitian and K"ahler structures and studies their behavior under T-duality in hyperk"ahler geometries.

Findings

Derived a T-duality covariant form of hyperk"ahler structures.

Established explicit T-duality relations between KK-monopole and H-monopole geometries.

Connected bi-hypercomplex structures with split-bi-quaternion algebras.

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 K\"{a}hler 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 K\"{a}hler structures are apparent. We also study the bi-hypercomplex and hyperk\"{a}hler cases in $\mathcal{N} = (4,4)$ theories. They are expressed, as a T-duality covariant fashion, in the generalized hyperk\"{a}hler structures and form the split-bi-quaternion algebras. As a concrete example, we show the explicit T-duality relation between the hyperk\"{a}hler 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.