Global Double Field Theory is Higher KaluzaKlein Theory
Abstract
KaluzaKlein Theory states that a metric on the total space of a principal bundle $P\rightarrow M$, if it is invariant under the principal action of $P$, naturally reduces to a metric together with a gauge field on the base manifold $M$. We propose a generalization of this KaluzaKlein principle to higher principal bundles and higher gauge fields. For the particular case of the abelian gerbe of KalbRamond field, this Higher KaluzaKlein geometry provides a natural global formulation for Double Field Theory (DFT). In this framework the doubled space is the total space of a higher principal bundle and the invariance under its higher principal action is exactly a global formulation of the familiar strong constraint. The patching problem of DFT is naturally solved by gluing the doubled space with a higher group of symmetries in a higher category. Locally we recover the familiar picture of an ordinary paraHermitian manifold equipped with Born geometry. Infinitesimally we recover the familiar picture of a higher Courant algebroid twisted by a gerbe (also known as Extended Riemannian Geometry). As first application we show that on a toruscompactified spacetime the Higher KaluzaKlein reduction gives automatically rise to abelian Tduality, while on a general principal bundle it gives rise to nonabelian Tduality. As final application we define a natural notion of Higher KaluzaKlein monopole by directly generalizing the ordinary GrossPerry one. Then we show that under Higher KaluzaKlein reduction, this monopole is exactly the NS5brane on a $10d$ spacetime. If, instead, we smear it along a compactified direction we recover the usual DFT monopole on a $9d$ spacetime.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.07089
 Bibcode:
 2019arXiv191207089A
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Differential Geometry
 EPrint:
 82 pages, 2 figures, 3 tables