We outline the construction of metastable de Sitter vacua of type IIB string theory. Our starting point is highly warped IIB compactifications with nontrivial NS and RR three-form fluxes. By incorporating known corrections to the superpotential from Euclidean D-brane instantons or gaugino condensation, one can make models with all moduli fixed, yielding a supersymmetric AdS vacuum. Inclusion of a small number of D3 branes in the resulting warped geometry allows one to uplift the AdS minimum and make it a metastable de Sitter ground state. The lifetime of our metastable de Sitter vacua is much greater than the cosmological timescale of 10 10 years. We also prove, under certain conditions, that the lifetime of dS space in string theory will always be shorter than the recurrence time.

We investigate the embedding of brane inflation into stable compactifications of string theory. At first sight a warped compactification geometry seems to produce a naturally flat inflaton potential, evading one well known difficulty of brane–antibrane scenarios. Careful consideration of the closed string moduli reveals a further obstacle: superpotential stabilization of the compactification volume typically modifies the inflaton potential and renders it too steep for inflation. We discuss the non-generic conditions under which this problem does not arise. We conclude that brane inflation models can only work if restrictive assumptions about the method of volume stabilization, the warping of the internal space, and the source of inflationary energy are satisfied. We argue that this may not be a real problem, given the large range of available fluxes and background geometries in string theory.

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.