Embedding tensor of Scherk-Schwarz flux compactifications from eleven dimensions

Abstract

We study the Scherk-Schwarz reduction of D = 11 supergravity with background fluxes in the context of a recently developed framework pertaining to D = 11 supergravity. We derive the embedding tensor of the associated four-dimensional maximal gauged theories directly from eleven dimensions by exploiting the generalised vielbein postulates, and by analysing the couplings of the full set of 56 electric and magnetic gauge fields to the generalised vielbeine. The treatment presented here will apply more generally to other reductions of D = 11 supergravity to maximal gauged theories in four dimensions.1 For a summary of recent developments and a complete bibliography see [8,9]. 2 In fact, the essential idea of reducing on a group manifold appears in [23]; for a useful historical account of Kaluza-Klein theory see [24].

Spheres, Generalised Parallelisability and Consistent Truncations

“…We see that in each case, provided the frame components of the fluxes and ∂ a φ and ∂ a ∆ are constant, then we are indeed in the class of generalised parallelisations with constant X AB C , that is we have a Leibniz generalised parallelisation. If we take ∂ a φ = f ab a = 0 or ∂ a ∆ = f ab a = 0 we see that these frame algebras match the standard gaugings in the literature [4,7,25] and [6,32]. We can also calculate the trace X A = X BA B .…”

“…The generalised Scherk-Schwarz ansatz (1.4) is also in practise used, for the metric components, in the original work on S 7 [30,9], and, recently, this has been extended to all the flux components [31]. In [32] the four-dimensional embedding tensor for conventional Scherk-Schwarz reductions was also calculated from eleven dimensions using the "generalised vielbein postulate" which, as we discuss in the conclusions, is connected to the algebra (1.3). The key point of this paper is to show that above conjecture also includes the sphere truncations.…”

Supersymmetric E7(7) exceptional field theory

“…These can be investigated along the lines of [40][41][42], exploiting the present formalism and the fact that it casts the higher-dimensional theory in a form adapted to (gauged) lower dimensional supergravity. Indeed, the full non-linear Kaluza-Klein ansätze for those higher-dimensional fields (including dual fields) yielding JHEP09(2014)044 scalar or vector fields in the compactification have already been obtained in this way for the AdS 4 ×S 7 compactification [6,[42][43][44], as well as for general Scherk-Schwarz compactifications with fluxes [45]. 3 Apart from the non-linear ansätze for higher rank tensors, which can now also be deduced in a straightforward fashion, and beyond the extension to other non-trivial compactifications of D = 11 supergravity, the main outstanding problem here is to extend these results to the compactification of IIB supergravity on AdS 5 × S 5 , for which either the supersymmetric extension of E 6(6) EFT [47] or the present version with the IIB solution of the section constraint might be employed.…”

“…4 The present reformulation naturally suggests that a higher-dimensional ancestor of the deformed SO(8) gauged supergravities might thus be obtained by performing an analogous 'twist' of the 56-bein of EFT (see also ref. [45]), V(x, Y ) → V(x, Y ; ω), relative to all vectors and tensors, where the 56-bein is now taken to also depend on the 56 extra coordinates Y M . Because of the inequivalence of the corresponding gauged SO (8) supergravities in four dimensions, it is clear that such a theory would no longer be onshell equivalent to the D = 11 supergravity of ref.…”

“…The GVPs are equations satisfied by the 56-bein and in the approach of [3,7,8] they can be checked explicitly on a component by component basis, while they appear as genuine postulates in the approach of the previous section. Moreover, the direct comparison with D = 11 supergravity allows for a direct understanding of four-dimensional maximal gauged theories and the embedding tensor [8,42,45] that defines them from a higher-dimensional perspective as well as providing generalized geometric structures that can be interpreted as generalized connections and used to construct a generalized curvature tensor.…”