Abstract
The truncation formulae of $D=11$ supergravity on $S^7$ to $D=4$ ${\cal N} =8$ SO(8)-gauged supergravity are completed to include the full non-linear dependence of the $D=11$ three-form potential $\hat A_{(3)}$ on the $D=4$ fields, and their consistency is shown. The full embedding into $\hat A_{(3)}$ is naturally expressed in terms of a restricted version, still ${\cal N} =8$ but only SL(8)--covariant, of the $D=4$ tensor hierarchy. The redundancies introduced by this approach are removed at the level of the field strength $\hat F_{(4)}$ by exploiting $D=4$ duality relations. Finally, new expressions for the full consistent truncation formulae are given that are, for the first time, explicit in all $D=4$ fields.
Highlights
D 1⁄4 11 supergravity [1] admits a consistent KaluzaKlein (KK) truncation on the seven-sphere to the purely electric SO(8) gauging [2] of maximal four-dimensional supergravity [3]
A formula for the combined dependence of the warp factor Δ and the inverse internal metric gmn on the S7 angles and the D 1⁄4 4 scalars has been known for a long time [4,5]
A similar formula has been given for the internal components Amnp of the D 1⁄4 11 three-form potential Að3Þ [7]. These formulas define the dependence of the warp factor
Summary
D 1⁄4 11 supergravity [1] admits a consistent KaluzaKlein (KK) truncation on the seven-sphere to the purely electric SO(8) gauging [2] of maximal four-dimensional supergravity [3]. The exact nonlinear dependence of the D 1⁄4 11 three-form potential on the fields of D 1⁄4 4 N 1⁄4 8 supergravity is computed, and the consistency of the corresponding embedding formulas is shown. This extends the proof of [4,7] of the truncation of D 1⁄4 11 supergravity on S7 at the level of the N 1⁄4 8 supersymmetry variations. M 1⁄4 1; ...; 56, i 1⁄4 1; ...; 8, of E7ð7Þ/SUð8Þ (through the square MMN 1⁄4 2VðMijVNÞij and through the gauge kinetic matrices I 1⁄2IJ1⁄2KL and R1⁄2IJ1⁄2KL); and of the embedding tensor ΘMα (through the vector field strengths and the scalar potential V). Using the embedding tensor formalism with (11), these field strengths can be computed to be HIð2JÞ 1⁄4 dAIJ − gδKLAIK ∧ ALJ; Hð2ÞIJ 1⁄4 dA IJ þ gδK1⁄2IAKL ∧ A JL þ 2gδK1⁄2IBJK; Hð3ÞI J
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