Special geometry of Euclidean supersymmetry IV: the local c-map

Abstract

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 Kähler.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 Kähler or para-Kähler base.

Non-extremal and non-BPS extremal five-dimensional black strings from generalized special real geometry

“…As we will show in separate publications [37,29], the full manifold M (3) is a para-quaternionic Kähler manifold. Here we restrict ourselves to investigating the geometry of the submanifold S. The manifold S is a totally geodesic submanifold of M (3) , since it is obtained by solving the equations of motion for 2n V + 1) degrees of freedom of the five-dimensional vector fields, see (7), or, equivalently, the corresponding three-dimensional scalars.…”

“…We then show that the manifold obtained by dimensional reduction to three dimensions takes the form S = N × Ê ⊂ M (3) , where N is a para-Kähler manifold that can be identified with the cotangent bundle T * M of a Hessian manifold M , which encodes the couplings of the original five-dimensional theory. While we restrict ourselves to the submanifold S relevant for black strings in this paper, the reduction of five-dimensional supergravity without and with vector multiplets to three Euclidean dimensions will be studied in depth in two companion papers [37,29].…”

Euclidean supergravity in five dimensions

“…A comprehensive study of four-dimensional supersymmetric theories with Euclidean spacetime signature has recently been conducted by Cortés, Mohaupt and collaborators in [1][2][3][4], where it is explained in detail how theories of N = 2 Euclidean 1 vector multiplets (both rigid and local) are constructed. In the case of Lorentzian signature it has been known for some time that the couplings of 4D, N = 2 vector multiplets are restricted by supersymmetry such that the scalar fields form a map into a target manifold with so-called special geometry [5].…”

“…Sabra), owen.vaughan@physics.org (O. Vaughan). 1 Following [1][2][3][4] we use the terminology N = 2 to refer to a supersymmetric theory with eight real supercharges regardless of spacetime dimension or signature. open problem.…”

Geroch group description of bubbling geometries

“…The further reduction to three dimensions leads to a Euclidean hypermultiplet theory: the symmetric target space (2.36) is symmetric pseudo-Riemannian para-quaternionic Kähler manifold -a special case of the general theory developed in references [30][31][32].…”