On the Flattening of Negative Curvature via T-Duality with a Non-Constant B-Field

TL;DR

This paper investigates whether T-duality with a non-constant B-field can flatten negative curvature spaces like AdS, concluding it is not possible to fully flatten AdS space through T-duality, despite certain transformations.

Contribution

It derives a relationship between curvature tensors under T-duality with B-fields and shows the limitations in flattening AdS space via T-duality.

Findings

T-duality cannot fully flatten AdS space regardless of B-field choice

A specific curvature component remains invariant under T-duality

Proposes a chain of dualities leading to a D9-brane geometry

Abstract

In an earlier paper, Alvarez, Alvarez-Gaume, Barbon and Lozano pointed out, that the only way to "flatten" negative curvature by means of a T-duality is by introducing an appropriate, non-constant NS-NS B-field. In this paper, we are investigating this further and ask, whether it is possible to T-dualize AdS_d space to flat space with some suitably chosen B. To answer this question, we derive a relationship between the original curvature tensor and the one of the T-dualized metric involving the B-field. It turns out that there is one particular component, which is independent of B. By inspection of this component, we then show, that it is not possible to dualize AdS_d to flat space irrespective of the choice of B. Finally, we examine the extension of AdS to an AdS_5 x S^5 geometry and propose a chain of S- and T-dualities together with an SL(2,Z) coordinate transformation, leading to a dual D9-brane geometry.