Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X_s.Comment: 14 pages, latex, one typo correcte

Homogeneous Lorentzian manifolds of a semisimple group

We describe the structure of $d$-dimensional homogeneous Lorentzian $G$-manifolds $M=G/H$ of a semisimple Lie group $G$. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group $G$ acts properly, that is the stabilizer $H$ is compact. Then any homogeneous space $G/\bar H$ with a smaller group $\bar H \subset H$ admits an invariant Lorentzian metric. A homogeneous manifold $G/H$ with a connected compact stabilizer $H$ is called a minimal admissible manifold if it admits an invariant Lorentzian metric, but no homogeneous $G$-manifold $G/\tilde H$ with a larger connected compact stabilizer $\tilde H \supset H$ admits such a metric. We give a description of minimal homogeneous Lorentzian $n$-dimensional $G$-manifolds $M = G/H$ of a simple (compact or noncompact) Lie group $G$. For $n \leq 11$, we obtain a list of all such manifolds $M$ and describe invariant Lorentzian metrics on $M$

Local reflexion spaces

A reflexion space is generalization of a symmetric space introduced by O. Loos. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.Comment: 8 pages, submitted to Archivum Mathematicu

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

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.Comment: 36 page

Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form \omega . This generalises Freed's definition of (affine) special K\"ahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and K\"ahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n. Such a realisation induces a canonical \nabla-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special K\"ahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms \alpha. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-K\"ahler structure on the cotangent bundle of a special K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and Introduction, version to appear in J. Geom. Phy

Compact Riemannian Manifolds with Homogeneous Geodesics

A homogeneous Riemannian space (M = G/H,g) is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group G. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric g with homogeneous geodesics on a homogeneous space of a compact Lie group G. We give a classification of compact simply connected GO-spaces (M = G/H,g) of positive Euler characteristic. If the group G is simple and the metric g does not come from a bi-invariant metric of G, then M is one of the flag manifolds M₁ = SO(2n+1)/U(n) or M₂ = Sp(n)/U(1)·Sp(n–1) and g is any invariant metric on M which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric g₀ such that (M,g0) is the symmetric space M = SO(2n+2)/U(n+1) or, respectively, CP²n⁻¹. The manifolds M₁, M₂ are weakly symmetric spaces

Supersymmetry reduction of N-extended supergravities in four dimensions

We consider the possible consistent truncation of N-extended supergravities to lower N' theories. The truncation, unlike the case of N-extended rigid theories, is non trivial and only in some cases it is sufficient just to delete the extra N-N' gravitino multiplets. We explore different cases (starting with N=8 down to N'\geq 2) where the reduction implies restrictions on the matter sector. We perform a detailed analysis of the interesting case N=2 \to N=1. This analysis finds applications in different contexts of superstring and M-theory dynamics.Comment: Version published on JHE

Completeness in supergravity constructions

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.Comment: 29 page

On paraquaternionic submersions between paraquaternionic K\"ahler manifolds

In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic K\"ahler non locally hyper paraK\"ahler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.Comment: 13 pages, no figure

D-term cosmic strings from N=2 Supergravity

We describe new half-BPS cosmic string solutions in N=2, d=4 supergravity coupled to one vector multiplet and one hypermultiplet. They are closely related to D-term strings in N=1 supergravity. Fields of the N=2 theory that are frozen in the solution contribute to the triplet moment map of the quaternionic isometries and leave their trace in N=1 as a constant Fayet-Iliopoulos term. The choice of U(1) gauging and of special geometry are crucial. The construction gives rise to a non-minimal Kaehler potential and can be generalized to higher dimensional quaternionic-Kaehler manifolds.Comment: 37 pages, LaTeX, v2: minor corrections, references added, version to be published in JHE