We found double-extreme black holes associated with the special geometry of the Calabi-Yau moduli space with the prepotential F = S T U .The area formula is S T U -moduli independent and has [SL (2; Z)] 3 symmetry in space of charges. The dual version of this theory without prepotential treats the dilaton S asymmetric versus T ; U -moduli. We display the dual relation between new (STU) black holes and stringy (SjT U ) black holes using particular S p (8; Z) transformation. The area formula of one theory equals the area formula of the dual theory when expressed in terms of dual charges.We analyse the relation between (STU)black holes to string triality of black holes: (SjT U ), (T jU S ), (UjST) solutions. In democratic S T U -symmetric version we nd that all three S and T and U duality symmetries are non-perturbative and mix electric and magnetic charges.

In Nϭ2 ungauged supergravity we have found the most general double-extreme dyonic black holes with arbitrary number n v of constant vector multiplets and n h of constant hypermultiplets. They are double extreme: ͑1͒ supersymmetric with coinciding horizons; ͑2͒ the mass for a given set of quantized charges is extremal. The spacetime is of the Reissner-Nordström form and the vector multiplet moduli depend on dyon charges. As an example we display n v complex moduli as functions of 2(n v ϩ1) electric and magnetic charges in a model related to a classical Calabi-Yau moduli space. A specific case includes the complex S, T, U moduli depending on four electric and four magnetic charges of 4 U͑1͒ gauge groups. ͓S0556-2821͑96͒04322-6͔

We study critical points of the BPS mass 2, the BPS string tension Z,, the black hole potential V and the gauged central charge potential P for M-theory compactified on Calabi-Yau three-folds. We first show that the stabilization equations for 2 (determining the black hole entropy) take an extremely simple form in five dimensions as opposed to four dimensions. The stabilization equations for Z, are also very simple and determine the size of the infinite a&33-throat of the string. The black hole potential in general exhibits two classes of critical points: supersymmetric critical points which coincide with those of the central charge and non-supersymmetric critical points. We then generalize the discussion to the entire extended Kahler cone encompassing topologically different but birationally equivalent Calabi-Yau threefolds that are connected via flop transitions. We examine behavior of the four potentials to probe the nature of these phase transitions. We find that V and P are continuous but not smooth across the flop transition, while 2 and its first two derivatives, as well as 2, and its first derivative, are continuous. This in turn implies that supersymmetric stabilization of 2 and 2, for a given configuration takes place in at most one point throughout the entire extended Kihler cone. The corresponding black holes (or string states) interpolate between different Calabi-Yau three-folds. At the boundaries of the extended KBhler cone we observe that electric states become massless and/or magnetic strings become tensionless.'aschouQslac.stanford.edu, kalloshQphysics.stanford.edu, rahmfeldQleland.stanford.edu, sjreyQsns.ias.edu, shmakovaQslac.stanford.edu, wkwongQleland.stanford.edu --'T.. 'It is to be noted that universal threshold corrections still leaves a possibility of coupling constant unification within weakly coupled heterotic string theory [4].