Kaluza-Klein consistency, Killing vectors and Kähler spaces

We make a detailed investigation of all spaces Q n 1 ··· n N q 1 ··· q N of the form of U (1) bundles over arbitrary products ∏ i CP n i of complex projective spaces, with arbitrary winding numbers q i over each factor in the base. Special cases, including Q 11 11 (sometimes known as T 11 ), Q 111 111 and Q 21 32 , are relevant for compactifications of type IIB and D = 11 supergravity. Remarkable `conspiracies' allow consistent Kaluza-Klein S 5 , S 4 and S 7 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 Q n 1 ··· n N q 1 ··· q N spaces, and that it is inconsistent to make a massless truncation in which the non-Abelian SU ( n i + 1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Q n 1 ··· n N q 1 ··· q N of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where q i = ( n i + 1)/ℓ all admit two Killing spinors. We also obtain an iterative construction for real metrics on CP n , and construct the Killing vectors on Q n 1 ··· n N q 1 ··· q N in terms of scalar eigenfunctions on CP n i . 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 Q n 1 ··· n N q 1 ··· q N .