Global Double Field Theory is Higher Kaluza-Klein Theory

Abstract

Kaluza-Klein Theory states that a metric on the total space of a principal bundle $P\to M$, 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 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.

arXiv comment83 pages, 2 figures, 3 tables. (Reference bug fixed.)

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Cited by in corpus (9)

• The Geometry, Branes and Applications of Exceptional Field Theory
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• On non-abelian U-duality of 11D backgrounds
• Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation
• Double Field Theory and Geometric Quantisation
• Cyclification of Orbifolds
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Gerbes in Geometry, Field Theory, and Quantisation