Kaluza-Klein Consistency, Killing Vectors, and Kahler Spaces

TL;DR

The paper investigates KK consistency for compactifications on a broad class of internal spaces $Q_{n_1 o n_N}^{q_1 o q_N}$, defined as $U(1)$ bundles over products of CP$^{n_i}$. By deriving a general necessary condition $Y^{IJ}$ for consistency in KK reductions, it shows that while sphere reductions satisfy this condition, the non-spherical bundle spaces do not, ruling out massless truncations that retain non-abelian $SU(n_i+1)$ gauge fields. The authors construct Killing vectors on these bundles from scalar eigenfunctions on CP$^n$, prove the existence of Einstein metrics for all nonzero winding data, and identify a supersymmetric subfamily with two Killing spinors when $q_i=(n_i+1)/ ext{ℓ}$. Collectively, these results imply that only the massless supergravity multiplet fields can be consistently retained in KK reductions on these spaces, with notable implications for AdS/CFT/Holography and the spectrum of holographic correlators.

Abstract

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 \prod_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)/\ell 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}.