The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective

TL;DR

This paper explores the global geometric structure of Double Field Theory (DFT), proposing a bundle gerbe framework with local charts that align with DFT coordinates, and discusses the implications for non-geometric string backgrounds.

Contribution

It introduces an atlas for the bundle gerbe in DFT, making the global geometric description more explicit and connecting local charts with standard DFT coordinates.

Findings

The atlas provides a natural local description of DFT.

Global non-geometric properties are clarified through tensor hierarchies.

The framework explains the emergence of non-abelian string bundles.

Abstract

The history of the geometry of Double Field Theory is the history of string theorists' effort to tame higher geometric structures. In this spirit, the first part of this paper will contain a brief overview on the literature of geometry of DFT, focusing on the attempts of a global description. In arXiv:1912.07089 [hep-th] we proposed that the global doubled space is not a manifold, but the total space of a bundle gerbe. This would mean that DFT is a field theory on a bundle gerbe, in analogy with ordinary Kaluza-Klein Theory being a field theory on a principal bundle. In this paper we make the original construction by arXiv:1912.07089 [hep-th] significantly more immediate. This is achieved by introducing an atlas for the bundle gerbe. This atlas is naturally equipped with $2d$-dimensional local charts, where $d$ is the dimension of physical spacetime. We argue that the local charts of this atlas should be identified with the usual coordinate description of DFT. In the last part we will discuss aspects of the global geometry of tensor hierarchies in this bundle gerbe picture. This allows to identify their global non-geometric properties and explain how the picture of non-abelian String-bundles emerges. We interpret the abelian T-fold and the Poisson-Lie T-fold as global tensor hierarchies.