N=2 Extremal Black Holes and Intersecting Branes

Using a simple hypothesis about the degrees of freedom of intersecting branes we find a microscopic counting argument that reproduces the entropy of a class of BPS black holes of type IIA string theory on general Calabi Yau three folds. 1 [email protected] Recently [1] the Bekenstein-Hawking area law for the entropy of black holes was given a statistical interpretation by counting the number of microstates in string theory configurations involving D-branes [2]. These results were first obtained in N = 4, 8 supergravity theories in four and five dimensions [1][3][4] and they were extended to special cases of N = 2 theories in [5]. In this paper we study black holes in more general N = 2 theories corresponding to type IIA string theory compactified on Calabi-Yau three folds (CY3) which can also be viewed as M-theory on CY3 × S1. In type IIA string theory on a generic Calabi Yau manifold there are 1 + h11 different electric charges corresponding to D-zerobrane charge and to D-twobrane charges associated to the different homology classes of the Calabi Yau, as well as their magnetic duals: D-sixbrane charge and D-fourbrane charges. We focus on the limit of large Calabi Yau size in which instanton corrections to the prepotential are small. We consider here black holes that carry zero brane charge plus arbitrary fourbrane charge (q0, p ) but they have zero two brane and six brane charges p = qa = 0. (q0, p ) are integers measuring the number of D-zerobranes and D-fourbranes. In this limit entropy is given in terms of the intersection numbers Cabc of the Calabi Yau. The classical horizon area entropy of supersymmetric black holes depends only on the charges and is proportional to the square of the value of the central charge at the horizon (in four dimensions) [6]. The central charge of the N = 2 supersymmetry algebra is a moduli dependent combination of the charges mentioned above and, through the BPS condition, is equal to the mass of the black hole (in Einstein metric). The scalar fields at the horizon take the values that minimize the central charge, or mass in Einstein metric, for fixed values of the black hole charges. The calculation of the entropy of a BPS charged black hole reduces to minimizing the central charge of the supersymmetry algebra with respect to the value of the scalar fields. These are the values that the scalar fields take at the horizon which are determined by the charges, we replace back these values in the expression for the central charge, we square it and then we get the entropy [6]. All we need to know is the expression of the central charge in terms of the charges. This is encoded by the prepotential. The prepotential has a leading term involving the intersection numbers Cabc then some subleading terms, one involving the second Chern class c2, another involving the Euler number and finally some instanton corrections. In the limit of big Calabi-Yau manifolds only the leading term survives. Since the entropy is given by the value of the central charge at the horizon we need the Calabi Yau to be big at the horizon. This in turn implies the following condition on the charges q ≫ p. This condition can be understood from the BPS mass formula since the zerobranes want to make the Calabi Yau big while the fourbranes want to make it smaller. It can be checked that the solutions for the moduli in terms of the charges imply that the sizes of the Calabi Yau are stabilized by the two opposite tendencies of the zerobranes and fourbranes [7]. Using this method the classical entropy formula was derived in [7]