Dimensional reduction along time offers a powerful way to study stationary solutions of 4D symmetric supergravity models via group-theoretical methods. We apply this approach systematically to extremal, BPS and non-BPS, spherically symmetric black holes, and obtain their "fake superpotential" W . The latter provides first order equations for the radial problem, governs the mass and entropy formula and gives the semi-classical approximation to the radial wave function. To achieve this goal, we note that the Noether charge for the radial evolution must lie in a certain Lagrangian submanifold of a nilpotent orbit of the 3D continuous duality group, and construct a suitable parametrization of this Lagrangian. For general non-BPS extremal black holes in N = 8 supergravity, W is obtained by solving a non-standard diagonalization problem, which reduces to a sextic polynomial in W 2 whose coefficients are SU(8) invariant functions of the central charges. By consistent truncation we obtain W for other supergravity models with a symmetric moduli space. In particular, for the one-modulus S 3 model, W 2 is given explicitely as the root of a cubic polynomial. The STU model is investigated in detail and the nilpotency of the Noether charge is checked on explicit solutions.

Motivated by applications to black hole physics and duality, we study the effect of higher derivative corrections on the dimensional reduction of four-dimensional Einstein, Einstein-Liouville and Einstein-Maxwell gravity to one direction, as appropriate for stationary, spherically symmetric solutions. We construct a field redefinition scheme such that the one-dimensional Lagrangian is corrected only by powers of first derivatives of the fields, eliminating spurious modes and providing a suitable starting point for quantization. We show that the Ehlers symmetry, broken by the leading R 2 corrections in Einstein-Liouville gravity, can be restored by including contributions of Taub-NUT instantons. Finally, we give a preliminary discussion of the duality between higher-derivative F-term corrections on the vector and hypermultiplet branches in N = 2 supergravity in four dimensions.

Stationary, spherically symmetric solutions of N = 2 supergravity in 3+1 dimensions have been shown to correspond to holomorphic curves on the twistor space of the quaternionic-Kähler space which arises in the dimensional reduction along the time direction. In this note, we generalize this result to the case of 1/4-BPS black holes in N = 4 supergravity, and show that they too can be lifted to holomorphic curves on a "twistor space" Z, obtained by fibering the Grassmannian F = SO(8)/U(4) over the moduli space in three-dimensions SO(8, n v + 2)/SO(8) × SO(n v + 2). This provides a kind of octonionic generalization of the standard constructions in quaternionic geometry, and may be useful for generalizing the known BPS black hole solutions, and finding new non-BPS extremal solutions.