Black Hole Entropy from Bps Closed String

By using BPS closed string, the entropy is calculated of the extremal five dimensional black hole consisting of Dirichlet onebranes, Dirichlet fivebranes and Kaluza-Klein momentum in the flat background approximation. In our formulations we consider two kinds of BPS closed strings with or without a winding number. In the former case heavy excitation modes of closed strings are used to derive the entropy. In the latter case we have no oscillator modes and consider collective motion of such massless closed strings. The entropy is given by the number of the ways how we divide the Kaluza-Klein momentum among the massless closed strings. In both cases the black hole entropy is the same as the Bekenstein-Hawking entropy. We argue that the collective modes of closed strings without winding is equivalent to a single closed string with winding. We propose that the two closed string pictures are connected with the open string pictures by the modular transformation.