Algebroid Structures on Para-Hermitian Manifolds

TL;DR

This paper constructs a global D-bracket on para-Hermitian manifolds, linking it to Courant algebroids and providing a geometric framework for Double Field Theory's algebraic structures.

Contribution

It introduces a global construction of the D-bracket on para-Hermitian manifolds and interprets DFT twists as deformations of these structures.

Findings

D-bracket can be expressed as a sum of Courant brackets under integrability.

Para-Hermitian geometry naturally models the extended space-time in DFT.

Bracket twists correspond to geometric deformations of para-Hermitian structures.

Abstract

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure.