We consider N = 2 supergravity in four dimensions, coupled to an arbitrary number of vector-and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that imply the second-order equations of motion. The first-order flow turns out to be driven by Hamilton's characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the flow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS first-order equations is obtained that corresponds to a different squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes.

We study some symmetry and integrability properties of four-dimensional Einstein-Maxwell gravity with nonvanishing cosmological constant in the presence of Killing vectors. First of all, we consider stationary spacetimes, which lead, after a timelike Kaluza-Klein reduction followed by a dualization of the two vector fields, to a three-dimensional nonlinear sigma model coupled to gravity, whose target space is a noncompact version of CP 2 with SU(2, 1) isometry group. It is shown that the potential for the scalars, that arises from the cosmological constant in four dimensions, breaks three of the eight SU(2, 1) symmetries, corresponding to the generalized Ehlers and the two Harrison transformations. This leaves a semidirect product of a one-dimensional Heisenberg group and a translation group R 2 as residual symmetry. We show that, under the additional assumptions that the three-dimensional manifold is conformal to a product space R × Σ, and all fields depend only on the coordinate along R, the equations of motion are integrable. This generalizes the results of Leigh et al. in arXiv:1403.6511 to the case where also electromagnetic fields are present. In the second part of the paper we consider the purely gravitational spacetime admitting a second Killing vector that commutes with the timelike one. We write down the resulting two-dimensional action and discuss its symmetries. If the fields depend only on one of the two coordinates, the equations of motion are again integrable, and the solution turns out to be one constructed by Krasiński many years ago.

We derive both BPS and non-BPS first-order flow equations for magnetically charged black strings in five-dimensional N = 2 abelian gauged supergravity, using the Hamilton-Jacobi formalism. This is first done for the coupling to vector multiplets only and U(1) Fayet-Iliopoulos (FI) gauging, and then generalized to the case where also hypermultiplets are present, and abelian symmetries of the quaternionic hyperscalar target space are gauged. We then use these results to derive the attractor equations for near-horizon geometries of extremal black strings, and solve them explicitely for the case where the constants appearing in the Chern-Simons term of the supergravity action satisfy an adjoint identity. This allows to compute in generality the central charge of the two-dimensional conformal field theory that describes the black strings in the infrared, in terms of the magnetic charges, the CY intersection numbers and the FI constants. Finally, we extend the r-map to gauged supergravity and use it to relate our flow equations to those in four dimensions.

We analytically construct asymptotically AdS 5 black string solutions starting from the four-dimensional domain wall black hole of [1]. It is shown that its uplift gives a black string in d = 5 minimal gauged supergravity, with momentum along the string. Applying instead the residual symmetries of N = 2, d = 4 Fayet-Iliopoulosgauged supergravity discovered in [2] to the domain wall seed leads, after uplifting, to a dyonic black string that interpolates between AdS 5 and AdS 3 × H 2 at the horizon. A Kaluza-Klein reduction of the latter along an angular Killing direction φ followed by a duality transformation yields, after going back to five dimensions, a black string with both momentum along the string and rotation along φ. This is the first instance of using solution-generating techniques in gauged supergravity to add rotation to a given seed. These solutions all have constant scalar fields. As was shown in [3], the construction of supersymmetric static magnetic black strings in the FI-gauged stu model amounts to solving the SO(2, 1) spinning top equations, which descend from an inhomogeneous version of the Nahm equations. We are able to solve these in a particular case, which leads to a generalization of the Maldacena-Nuñez solution.

We propose a geometric method to study the residual symmetries in N = 2, d = 4 U(1) Fayet-Iliopoulos (FI) gauged supergravity. It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory. In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed. We illustrate the procedure for finding this group in general and then show how it works in some specific models. For the prepotential F = −iX 0 X 1 , we use our method to add one more parameter to the rotating Chow-Compère solution, representing scalar hair.