A Comment on Quantum Distribution Functions and the OSV Conjecture

Using the attractor mechanism and the relation between the quantization of $H^{3}(M)$ and topological strings on a Calabi Yau threefold $M$ we define a map from BPS black holes into coherent states. This map allows us to represent the Bekenstein-Hawking-Wald entropy as a quantum distribution function on the phase space $H^{3}(M)$. This distribution function is a mixed Husimi/anti-Husimi distribution corresponding to the different normal ordering prescriptions for the string coupling and deviations of the complex structure moduli. From the integral representation of this distribution function in terms of the Wigner distribution we recover the Ooguri-Strominger-Vafa (OSV) conjecture in the region "at infinity" of the complex structure moduli space. The physical meaning of the OSV corrections are briefly discussed in this limit.