The Puzzle of Global Double Field Theory: Open Problems and the Case for a Higher Kaluza‐Klein Perspective

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 [1] 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 [1] 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.

Double Field Theory and Geometric Quantisation

“…(see [34] and [35] for details). By using the properties of the transgression functor from M to its loop space LM , we immediately obtain the new patching conditions…”

“…Therefore, the requirement that X (σ) is periodic can be immediately recasted as the requirement that the pullback of dx M is periodic. References [34] and [35] explore the idea that the doubled space M is globally not a smooth manifold, but a more generalised geometric object (see there for details). In particular, in the references, it is derived that the patching conditions for local coordinate patches U (α) and U (β) of the doubled space M should be of the form…”

“…Since the Čech cocycle condition dΛ M (αβ) + dΛ M (βγ) + dΛ M (γα) = 0 is satisfied, we do not encounter problems for X * dx M = dX(σ) being periodic. In other words, a doubled string can naturally live on a doubled space M that is patched in a more general way than a manifold (like the proposal by [34] and [35]) exactly because X(σ) does not appear in the action, but only X (σ) does.…”

Double field theory and geometric quantisation

“…We remark that, in this more general setting, the doubled space M is not required to be a product space. References [51] and [52] explore the idea that a general doubled space M is globally not a smooth manifold, but a more generalised geometric object (see there for details). In particular, in the references, it is derived that the patching conditions for local coordinate patches U (γα) = 0 is satisfied, we do not encounter problems for X * dx M = dX(σ) being periodic.…”

“…In particular, in the references, it is derived that the patching conditions for local coordinate patches U (γα) = 0 is satisfied, we do not encounter problems for X * dx M = dX(σ) being periodic. In other words, a doubled string can naturally live on a doubled space M that is patched in a more general way than a manifold (like the proposal by [51] and [52]) exactly because X(σ) does not appear in the action, but only X (σ) does.…”