Geometry of the embedding of supergravity scalar manifolds in D=11 and D=10
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Several recent papers have made considerable progress in proving the existence of remarkable consistent Kaluza-Klein sphere reductions of D=10 and D=11 supergravities, to give gauged supergravities in lower dimensions. A proof of the consistency of the full gauged SO(8) reduction on S7 from D=11 was given many years ago, but from a practical viewpoint a reduction to a smaller subset of the fields can be more manageable and explicit, for the purposes of lifting lower-dimensional solutions back to the higher dimension. The major complexity of the spherical reduction Anstze comes from the spin-0 fields, and of these, it is the pseudoscalars that are the most difficult to handle. In this paper we address this problem in two cases. One arises in a truncation of SO(8) gauged supergravity in four dimensions to U(1)4, where there are three pairs of dilatons and axions in the scalar sector. The other example involves the truncation of SO(6) gauged supergravity in D=5 to a subsector containing a scalar and a pseudoscalar field, with a potential that admits a second supersymmetric vacuum aside from the maximally-supersymmetric one. We briefly discuss the use of these embedding anstze for the lifting of solutions back to the higher dimension. 2000 Elsevier Science B.V.