Generalised geometry from the ground up

Abstract

Extending previous work on generalised geometry, we explicitly construct an E 7(7) -valued vielbein in eleven dimensions that encompasses the would be scalar bosonic degrees of freedom after reduction to four dimensions of D = 11 supergravity, by identifying new "generalised vielbeine" in eleven dimensions associated with the dual 6-form potential and the dual graviton. By maintaining full on-shell equivalence with the original theory at every step our construction furnishes the non-linear ansatz for the dual (magnetic) 7-form flux for any non-trivial compactification of D = 11 supergravity, complementing the known non-linear ansätze for the metric and the 4-form flux. A preliminary analysis of the generalised vielbein postulate for the new vielbein components reveals tantalising hints of new structures beyond D = 11 supergravity and ordinary space-time covariancxe, and also points to the possible D = 11 origins of the embedding tensor. We discuss the extension of these results to E 8(8) .

Embedding of gauged STU supergravity in eleven dimensions

“…In that paper, the complete expression for the eleven-dimensional metric ansatz was obtained, and partial expressions also for the four-form field strength. The complete expressions for the uplifted four-form field strength were obtained recently [13][14][15][16]. Although the results are now complete, they are not necessarily convenient to use in practice, if, as is commonly the case, one is interested in uplifting four-dimensional configurations that involve only a subset of the full set of N = 8 supergravity fields.…”

“…Their construction, making extensive use of the supersymmetry of the theory, essentially proves that the reduction is necessarily a consistent one. Much more recently, further work has provided more complete expressions for the 4-form uplift ansatz, and has also provided further insights into the structure of the reduction [13][14][15][16].…”

Supersymmetric E7(7) exceptional field theory

“…[20,21]. The purpose of this paper, then, is to bring together these strands of development: first we complete the construction of the E 7(7) EFT by giving the fully supersymmetric extension by fermions; second, we relate the resulting theory to the formulation of [3,[6][7][8]. As one of our main results we will demonstrate the compatibility of these two formulations, and explain the subtleties involved in making a detailed comparison.…”

“…The approach of [3], which has been extended and completed in [7,8] to also take into account aspects of the E 7(7) -based exceptional geometry, takes D = 11 supergravity as the starting point and reformulates it in order to make a local SO(1, 3) × SU(8) tangent space symmetry manifest. To this end the fields and coordinates are decomposed in a (4 + 7) splitting, as in Kaluza-Klein compactifications, but keeping the full coordinate dependence of all fields (however, unlike in EFT, no extra coordinates beyond those of the original theory are introduced).…”

“…Moreover, those parts of the bosonic sector which lead to scalar and vector fields in the dimensionally reduced maximal supergravity can then be assembled into E 7(7) objects, namely a 56-bein encoding the internal field components and a 56-plet of vectors combining the 28 electric and 28 magnetic vectors of N = 8 supergravity; their supersymmetry transformations can be shown to take the precise form of the four-dimensional maximal gauged supergravity. While in this approach the fermions are included from the beginning (with the supersymmetry variations constituting the starting point of the analysis) and the on-shell equivalence with D = 11 supergravity is thus guaranteed at each step of the construction, a proper understanding of the role of E 7 (7) in eleven dimensions (as well as of the E 7(7) -covariant dynamics of the bosonic sector) was lacking in the original work of [3], and has only emerged with the recent advances. Nevertheless it is remarkable that the combinations of SU (8) connections in the supersymmetry variations of the fermions found 'empirically' in ref.…”