It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets XI with a generic holomorphic prepotential F(x.') can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the X" are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the Kahler potential on the vector multiplet moduli space. Several examples are discussed in detail.

We find a general principle which allows one to compute the area of the horizon of Nϭ2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space ͑a minimum corresponds to a fixed point of attraction͒. The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for Nϭ2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit modelindependent expression for the macroscopic Bekenstein-Hawking entropy of Nϭ2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in Nϭ4,8 supersymmetries and the relation to the Nϭ2 case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge. ͓S0556-2821͑96͒03714-9͔

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N≥2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for study of critical phenomena.

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The general form of N = 2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Lagrangians for rigid theories are also written in this general setting and the connection with local theories elucidated. The derivation of these results using geometrical techniques is briefly summarized.

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We study general properties of the low-energy effective theory for 4D type II superstrings obtained by the compactification on an abstract (2,2) superconformal system. This is the basic step towards the construction of their moduli space. We give an explicit and general algorithm to convert the effective Lagrangian for the type IIA into that of type IIB superstring defined by the same (2,2) superconformal system (and vice versa). This map converts Kahler manifolds into quaternionic ones (and quaternionic into Kahlerian ones) and has a deep geometrical meaning. The relationship with the theory of normal quaternionic manifolds (and algebras), as well as with Jordan algebras, is outlined. It turns out that only a restricted class of quarternionic geometries is allowed in the string case. We derive a general and explicit formula for the (fully nonlinear) couplings of the vector-multiplets (IIA case) in terms of the basic three-point functions of the underlying superconformal theory. A number of illustrative examples is also presented.

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