Kaluza-Klein consistency, Killing vectors and Kähler spaces

We make a detailed investigation of all spaces Qn1nNq1qN of the form of U(1) bundles over arbitrary products ∏iCPni of complex projective spaces, with arbitrary winding numbers qi over each factor in the base. Special cases, including Q1111 (sometimes known as T11), Q111111 and Q2132, are relevant for compactifications of type IIB and D = 11 supergravity. Remarkable `conspiracies' allow consistent Kaluza-Klein S5, S4 and S7 sphere reductions of these theories that retain all the Yang-Mills fields of the isometry group in a massless truncation. We prove that such conspiracies do not occur for the reductions on the Qn1nNq1qN spaces, and that it is inconsistent to make a massless truncation in which the non-Abelian SU(ni + 1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Qn1nNq1qN of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where qi = (ni + 1)/l all admit two Killing spinors. We also obtain an iterative construction for real metrics on CPn, and construct the Killing vectors on Qn1nNq1qN in terms of scalar eigenfunctions on CPni. We derive bounds that allow us to prove that certain Killing-vector identities on spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied on Qn1nNq1qN.