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Abstract: In this Note we discuss the application of the Hamilton–Jacobi formalism to the first order description of four-dimensional spherically symmetric and static black holes. In particular we show that the prepotential characterizing the flow coincides with the Hamilton principal function associated with the one-dimensional effective Lagrangian. This implies that the prepotential can always be defined, at least locally in the radial variable and in the moduli space, both in the extremal and non-extremal cases and allows us to conclude that it is duality invariant. We also give, in this framework, a general definition of the “Weinhold metric” in terms of which a necessary condition for the existence of multiple attractors is given. The Hamilton–Jacobi formalism can be applied both to the restricted phase space where the electromagnetic potentials have been integrated out as well as in the case where the electromagnetic potentials are dualized to scalar fields using the so-called three-dimensional Euclidean approach. We give some examples of application of the formalism, both for the BPS and the non-BPS black holes. [Copyright &y& Elsevier]

Abstract: We consider the attractor equations of particular , supergravity models whose vector multiplets'' scalar manifold is endowed with homogeneous symmetric cubic special Kähler geometry, namely of the so-called and stu models. In this framework, we derive explicit expressions for the critical moduli corresponding to non-BPS attractors with vanishing central charge. Such formulæ hold for a generic black hole charge configuration, and they are obtained without formulating any ad hoc simplifying assumption. We find that such attractors are related to the -BPS ones by complex conjugation of some moduli. By uplifting to , supergravity, we give an interpretation of such a relation as an exchange of two of the four eigenvalues of the central charge matrix . We also consider non-BPS attractors with non-vanishing ; for peculiar charge configurations, we derive solutions violating the ansatz usually formulated in literature. Finally, by group-theoretical considerations we relate Cayley''s hyperdeterminant (the invariant of the stu model) to the invariants of the and of the so-called model. [Copyright &y& Elsevier]

Abstract: We constructed the pp-wave limit of superconformal mechanics with the off-shell multiplet. We present the superfield and the component actions which exhibit the interesting property that the interaction parts are completely fixed by the symmetry. We also explicitly demonstrate that the passing to the pp-wave limit can be achieved by keeping at most quadratic nonlinearities in the action of (super)conformal mechanics. [Copyright &y& Elsevier]

Abstract: We construct supersymmetric mechanics using the nonlinear chiral supermultiplet. The two bosonic degrees of freedom of this supermultiplet parameterize the sphere and go into the bosonic components of the standard chiral multiplet when the radius of the sphere goes to infinity. We construct the most general action and demonstrate that the nonlinearity of the supermultiplet results in the deformation of the connection, which couples the fermionic degrees of freedom with the background, and of the bosonic potential. Also a non-zero magnetic field could appear in the system. [Copyright &y& Elsevier]