On the Riemann Tensor in Double Field Theory

TL;DR

This paper develops a duality covariant Riemann tensor within double field theory, extending geometric concepts to include T-duality invariance, and explores implications for string theory effective actions and ' corrections.

Contribution

It introduces a duality covariant Riemann tensor in double field theory and investigates its properties and limitations in relation to T-duality and higher-derivative corrections.

Findings

The duality covariant Riemann tensor's contractions yield Ricci and scalar curvatures.

No T-duality invariant four-derivative object reduces to the square of the Riemann tensor.

' corrections likely require modifications to T-duality transformations.

Abstract

Double field theory provides T-duality covariant generalized tensors that are natural extensions of the scalar and Ricci curvatures of Riemannian geometry. We search for a similar extension of the Riemann curvature tensor by developing a geometry based on the generalized metric and the dilaton. We find a duality covariant Riemann tensor whose contractions give the Ricci and scalar curvatures, but that is not fully determined in terms of the physical fields. This suggests that \alpha' corrections to the effective action require \alpha' corrections to T-duality transformations and/or generalized diffeomorphisms. Further evidence to this effect is found by an additional computation that shows that there is no T-duality invariant four-derivative object built from the generalized metric and the dilaton that reduces to the square of the Riemann tensor.