Global Double Field Theory is Higher Kaluza‐Klein Theory

Abstract

AbstractKaluza‐Klein Theory states that a metric on the total space of a principal bundle , 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 Kaluza‐Klein principle to higher principal bundles and higher gauge fields. For the particular case of the abelian gerbe of Kalb‐Ramond field, this Higher Kaluza‐Klein 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 para‐Hermitian 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 torus‐compactified spacetime the Higher Kaluza‐Klein reduction gives automatically rise to abelian T‐duality, while on a general principal bundle it gives rise to non‐abelian T‐duality. As final application we define a natural notion of Higher Kaluza‐Klein monopole by directly generalizing the ordinary Gross‐Perry one. Then we show that under Higher Kaluza‐Klein reduction, this monopole is exactly the NS5‐brane on a 10d spacetime. If, instead, we smear it along a compactified direction we recover the usual DFT monopole on a 9d spacetime.