T-duality on Almost Hermitian Spaces

TL;DR

This work examines Abelian T-duality on almost bi-hermitian spaces with torsion, deriving full transformation rules for $g$, $B$, $\phi$ and for the almost complex structures $J_{\pm}$ and their forms $\omega_{\pm}$ via the Buscher rule. It then analyzes how integrability, expressed by the vanishing of the Nijenhuis tensors, behaves under duality in three settings: bi-hermitian with torsion, Hermitian without torsion, and Kähler; the key finding is that integrability is generally not preserved, but Kähler initial data do preserve integrability on the dual, with a dual theory that can host H-flux and non-Kähler geometry. The paper provides explicit, worked examples—CP$^2$ and Taub-NUT—showing that CP$^2$ duals remain complex with covariantly constant dual $J'_{\pm}$ but non-closed $\omega'_{\pm}$, while Taub-NUT duals yield an H-monopole with integrable, commuting triples $J'_{a,\±}$ and corresponding $\omega'_{a,\±}$. These results clarify how T-duality interplays with complex geometry in string compactifications and illuminate connections to generalized geometry and flux backgrounds.

Abstract

We investigate T-duality transformation on an almost bi-hermitian space with torsion. By virtue of the Buscher rule, we completely describe not only the covariant derivative of geometrical objects but also the Nijenhuis tensor. We apply this description to an almost bi-hermitian space with isometry and investigate integrability on its T-dualized one. We find that hermiticity is not a sufficient condition to preserve integrability under T-duality transformations. However, in the presence of the Kähler condition, the T-dualized space still admits integrability of the almost complex structures. We also observe that the form of H-flux is suitable for string compactification scenarios.