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

TL;DR

This paper demonstrates that certain quotients of quaternionic K"ahler manifolds, arising from specific group actions, naturally inherit K"ahler structures, with implications for supergravity and supersymmetry breaking.

Contribution

It proves the existence of natural K"ahler structures on quotients of quaternionic K"ahler manifolds under specific group actions, especially in the context of the supergravity c-map.

Findings

Quotients M' = N/A have natural K"ahler structures.

All quaternionic K"ahler manifolds from the c-map admit compatible complex structures.

The K"ahler structure is essential for supersymmetry breaking consistency.

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.