Consistent Warped-Space Kaluza-Klein Reductions, Half-Maximal Gauged Supergravities and CP^n Constructions

TL;DR

The paper addresses constructing consistent warped Kaluza-Klein embeddings of half-maximal gauged supergravities in $D=7$, $D=6$, $D=5$ and $D=4$ by performing circle reductions or T-duality on Hopf fibres of $S^3$ inside known sphere reductions, yielding internal spaces that act as mirror manifolds to the original spheres.The authors obtain explicit warped embeddings across IIA, IIB, M-theory, and relate AdS vacua of the lower-dimensional theories to warped AdS spacetimes with these internal spaces; they also present non-singular warped embeddings for $D=5$ and $D=4$ with CP$^n$ internal geometries and examine the associated Fubini-Study metrics.The work broadens the landscape of consistent reductions for half-maximal gauged supergravities, highlighting when warp factors are singular or non-singular and detailing the emergence of CP$^n$ structures, with potential implications for AdS/CFT contexts and higher-dimensional uplifts.Overall, the paper provides a systematic geometric construction that connects AdS$_{d}$ vacua to warped higher-dimensional embeddings and clarifies the role of complex projective spaces in the internal geometry of these reductions.

Abstract

We obtain new consistent Kaluza-Klein embeddings of the gauged supergravities with half of maximal supersymmetry in dimensions D=7, 6, 5 and 4. They take the form of warped embeddings in type IIA, type IIB, M-theory and type IIB respectively, and are obtained by performing Kaluza-Klein circle reductions or T-duality transformations on Hopf fibres in S^3 submanifolds of the previously-known sphere reductions. The new internal spaces are in some sense ``mirror manifolds'' that are dual to the original internal spheres. The vacuum AdS solutions of the gauged supergravities then give rise to warped products with these internal spaces. As well as these embeddings, which have singularities, we also construct new non-singular warped Kaluza-Klein embeddings for the D=5 and D=4 gauged supergravities. The geometry of the internal spaces in these cases leads us to study Fubini-Study metrics on complex projective spaces in some detail.