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 → 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.

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

“…Notice that, in the particular case of a tensor hierarchy where all the fields do not depend on the internal space , the curvature reduces to the familiar equations of a doubled torus bundle, i.e. which is exactly the curvature of the ‐bundle araising in the case of a globally geometric T‐duality, as explained in [1]. Also the gauge transformations reduce to The local field can thus be globalized to the curvature of a doubled torus bundle with 1st Chern class .…”

“…As argued in [6] and [3], DFT should be interpreted as a generalization of Kaluza‐Klein Theory where it is the Kalb‐Ramond field, and not a gauge field, that is unified with the pseudo‐Riemannian metric in a bigger space. Since the Kalb‐Ramond field is geometrized by a bundle gerbe, in [1] 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. In the reference we showed how to derive some known doubled spaces such as the ones describing T‐folds, and how to interpret T‐duality.…”

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.…”

Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

“…An extension similar to (1.2) was considered in [FRS16], where symmetries of a gerbe with connection were investigated in relation with higher geometric prequantisation. Infinitesimal versions of the extension (1.2) were considered in [Col11,FRS16], where it was shown that these give rise to the standard H -twisted Courant algebroid on M, where H is the 3-form curvature of the connection on G. These considerations have been expanded on and applied to higher versions of Kaluza-Klein reductions of string theory in [Alf20].…”