Fractional branes and the entropy of 4D black holes

We reconsider the four dimensional extremal black hole constructed in type- IIB string theory as the bound state of D1-branes, D5-branes, momentum, and Kaluza-Klein monopoles. Specifically, we examine the case of an arbitrary number of monopoles. Consequently, the weak coupling calculation of the mic roscopic entropy requires a study of the D1-D5 system on an ALE space. We f ind that the complete expression for the Bekenstein-Hawking entropy is obta ined by taking into account the massless open strings stretched between the fractional D-branes which arise in the orbifold limit of the ALE space. Th e black hole sector therefore arises as a mixed Higgs-Coulomb branch of an effective 1 + 1 dimensional gauge theory.