We consider the geodesic motion on the symmetric moduli spaces that arise after timelike and spacelike reductions of supergravity theories. The geodesics correspond to timelike respectively spacelike $p$-brane solutions when they are lifted over a $p
$-dimensional flat space. In particular, we consider the problem of constructing emph{the minimal generating solution}: A geodesic with the minimal number of free parameters such that all other geodesics are generated through isometries. We give an intrinsic characterization of this solution in a wide class of orbits for various supergravities in different dimensions. We apply our method to three cases: (i) Einstein vacuum solutions, (ii) extreme and non-extreme D=4 black holes in N=8 supergravity and their relation to N=2 STU black holes and (iii) Euclidean wormholes in $Dgeq 3$. In case (iii) we present an easy and general criterium for the existence of regular wormholes for a given scalar coset.
We study N=2 supergravity deformed by a genuine supersymmetric completion of the $lambda R^4$ term, using the underlying off shell N=2 superconformal framework. The gauge-fixed superconformal model has unbroken local supersymmetry of N=2 supergravity
with higher derivative deformation. Elimination of auxiliary fields leads to the deformation of the supersymmetry rules as well as to the deformation of the action, which becomes a Born-Infeld with higher derivative type action. We find that the gravitino supersymmetry deformation starts from $lambda , pa^4 {cal F}^3$ and has higher graviphoton couplings. In the action there are terms $lambda^2 pa^8 {cal F}^{6}$ and higher, in addition to original on shell counterterm deformation. These deformations are absent in the on shell superspace and in the candidate on shell counterterms of N=4,~8 supergravities, truncated down to N=2. We conclude therefore that the undeformed on shell superspace candidate counterterms break the N=2 part of local supersymmetry.
In this paper we continue the program of the classification of nilpotent orbits using the approach developed in arXiv:1107.5986, within the study of black hole solutions in D=4 supergravities. Our goal in this work is to classify static, single cente
r black hole solutions to a specific N=2 four dimensional magic model, with special Kahler scalar manifold ${rm Sp}(6,mathbb{R})/{rm U}(3)$, as orbits of geodesics on the pseudo-quaternionic manifold ${rm F}_{4(4)}/[{rm SL}(2,mathbb{R})times {rm Sp}(6,mathbb{R})]$ with respect to the action of the isometry group ${rm F}_{4(4)}$. Our analysis amounts to the classification of the orbits of the geodesic velocity vector with respect to the isotropy group $H^*={rm SL}(2,mathbb{R})times {rm Sp}(6,mathbb{R})$, which include a thorough classification of the emph{nilpotent orbits} associated with extremal solutions and reveals a richer structure than the one predicted by the $beta-gamma$ labels alone, based on the Kostant Sekiguchi approach. We provide a general proof of the conjecture made in arXiv:0908.1742 which states that regular single center solutions belong to orbits with coinciding $beta-gamma$ labels. We also prove that the reverse is not true by finding distinct orbits with the same $beta-gamma$ labels, which are distinguished by suitably devised tensor classifiers. Only one of these is generated by regular solutions. Since regular static solutions only occur with nilpotent degree not exceeding 3, we only discuss representatives of these orbits in terms of black hole solutions. We prove that these representatives can be found in the form of a purely dilatonic four-charge solution (the generating solution in D=3) and this allows us to identify the orbit corresponding to the regular four-dimensional metrics.
We review the status of the integrability and solvability of the geodesics equations of motion on symmetric coset spaces that appear as sigma models of supergravity theories when reduced over respectively the timelike and spacelike direction. Such ge
odesic curves describe respectively timelike and spacelike brane solutions. We emphasize the applications to black holes.
