Abstract
Consistent embeddings are found of the minimal mathcal{N} = 2 and mathcal{N} = 3 gauged supergravities in four dimensions into its maximally supersymmetric, mathcal{N} = 8, counterpart with a dyonic ISO(7) gauging. These minimal truncations retain the metric along with relevant U(l) and S0(3) R-symmetry gauge fields selected from the ISO (7) ones. The remaining ISO (7) gauge fields are turned off, with subtleties introduced by the dyonic gauging, and the scalars are fixed to their expectation values at the mathcal{N} = 2 and mathcal{N} = 3 vacua of theN = 8 theory. Using the truncation formulae for massive type I IA supergravity on the six-sphere to D = 4 mathcal{N} = 8 ISO (7) supergravity, the minimal D = 4 mathcal{N} = 2 and mathcal{N} = 3 gauged supergravities are then uplifted consistently to ten dimensions.
Highlights
A natural strategy to build consistent truncations of string or M-theory down to pure gauged supergravities relies on the existence of a G-structure description [28] of the background geometry
The N = 2 [21] and N = 3 [22, 23] AdS4 solutions of massive IIA arise as the S6 uplifts of critical points of ISO(7) supergravity that break the N = 8 supersymmetry of the theory to N = 2 [21] and N = 3 [52]
Some of the gauge fields that need to be truncated out are the dyonically gauged non-compact ones of ISO(7). This is done by writing them in terms of the surviving compact R-symmetry gauge fields and their Hodge duals, rather than by setting them to zero. Bring these field restrictions to the D = 10 to D = 4 consistent truncation formulae of [21, 44], in order to find the embedding into massive type IIA
Summary
Bringing the vevs (2.24) and the vector identifications (3.17) to (2.9), all the covariant derivatives are seen to vanish, except the ones of the Stuckelberg scalars aij, which produce the relation (3.1) The latter can be solved by letting A(t)i = 0, A(it) = 0, but the equations of motion set F i = 0. For further reassurance, the consistency of these minimal truncations is manifestly checked in appendix B at the level of the Bianchi identities and equations of motion of the type IIA supergravity forms. The D = 4 metric and graviphoton here are generic, though: they are only required to obey the D = 4 N = 2 field equations (A.1) Proceeding along these lines, some calculation gives the following formulae for the consistent truncation of massive IIA supergravity to minimal D = 4 N = 2 gauged supergravity (A.2): ds210.
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