On the geometry of double field theory

TL;DR

This paper reformulates double field theory using differential geometry, introducing metric algebroids and para-Dirac structures to better understand its geometric foundations related to T-duality.

Contribution

It expresses double field theory in invariant geometric terms, defining metric algebroids and para-Dirac structures, and constructs canonical connections with scalar curvatures for action principles.

Findings

Defined metric algebroids with Courant-like brackets

Constructed two canonical connections with scalar curvatures

Introduced para-Dirac structures on double manifolds

Abstract

Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in the framework of para-Kaehler manifolds. We define metric algebroids, which are vector bundles with a bracket of cross sections that has the same metric compatibility property as a Courant bracket. We show that a double field gives rise to two canonical connections, whose scalar curvatures can be integrated to obtain actions. Finally, in analogy with Dirac structures, we define and study para-Dirac structures on double manifolds.