Special Geometry of Euclidean Supersymmetry IV: the local c-map

TL;DR

This paper explores the geometric structures arising from different dimensional reductions of 4D N=2 supergravity theories, establishing new properties of the resulting scalar manifolds and providing a unified global construction of the c-maps.

Contribution

It introduces a comprehensive analysis of spatial, temporal, and Euclidean c-maps, proving para-quaternionic Kahler structures and constructing integrable complex and para-complex structures on the target manifolds.

Findings

Target manifolds are para-quaternionic Kahler in temporal and Euclidean reductions.

Constructed integrable complex structures on the target manifolds.

Provided a new global construction of the c-maps as fiber bundles over special Kahler bases.

Abstract

We consider timelike and spacelike reductions of 4D, N = 2 Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map, (ii) the temporal c-map, which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map, which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kahler. In cases (i) and (ii) we construct two integrable complex structures on the target manifold, one of which belongs to the quaternionic and para-quaternionic structure, respectively. In case (iii) we construct two integrable para-complex structures, one of which belongs to the para-quaternionic structure. In addition we provide a new global construction of the spatial, temporal and Euclidean c-maps, and separately consider a description of the target manifold as a fibre bundle over a projective special Kahler or para-Kahler base.