Abstract
AbstractThe 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.
Highlights
Introduction to Higher KaluzaKlein Theory we will give a very brief introduction to the Higher Kaluza-Klein perspective on the geometry of Double Field Theory (DFT) we started to develop in [Alf20].3.1 The doubled space as a bundle gerbeIn the Higher Kaluza-Klein proposal the doubled space of DFT is identified with the total space of a bundle gerbe with connection
Since the Kalb-Ramond field is geometrized by a bundle gerbe, in [Alf20] we proposed that DFT should be globally interpreted as a field theory on the total space of a bundle gerbe, just like ordinary Kaluza-Klein Theory lives on the total space of a principal bundle
Double Field Theory on group manifolds, known as DFTWZW, gives us a well-defined global version of DFT in the particular case of constant generalized fluxes, which can be interpreted as the structure constants of some 2d-dimensional Lie algebra
Summary
As explained by [BB20], Double Field Theory (DFT) should be thought as a generalization of Kaluza-Klein Theory from gauge fields to the Kalb-Ramond field. We want a tensor parametrizing the coset O(d, d)/ O(1, d − 1) × O(1, d − 1) , we will define the generalized metric GMN dxM ⊗ dxN by requiring that it is symmetric and it satisfies the property GMLηLP GP N = ηMN. Dxν is an anti-symmetric tensor on the submanifold U These are respectively interpreted as a metric and a Kalb-Ramond field on the d-dimensional patch U. The globalization problem of DFT can be summed as follows: since the Kalb-Ramond field Bμν is the locally defined 2-form of the connection of a bundle gerbe (see patching conditions (1.0.1)), how can local DFT patches U, GMN , ηMN we just introduced be glued together consistently? The globalization problem of DFT can be summed as follows: since the Kalb-Ramond field Bμν is the locally defined 2-form of the connection of a bundle gerbe (see patching conditions (1.0.1)), how can local DFT patches U, GMN , ηMN we just introduced be glued together consistently? We will devolve the rest of the paper to try to answer this question
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