On certain K\"ahler quotients of quaternionic K\"ahler manifolds

Abstract

We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic K\"ahler manifold M which preserves a submanifold N\subset M, the quotient M'=N/A has a natural K\"ahler structure. We verify that the assumptions on the group action and on the submanifold N\subset M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic K\"ahler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N\subset M is a complex submanifold. Finally, we discuss how the existence of the K\"ahler structure on M' is required by the consistency of spontaneous {\cal N}=2 to {\cal N}=1 supersymmetry breaking.