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 amap from BPS black holes into coherent states. This map allows us to representthe Bekenstein-Hawking-Wald entropy as a quantum distribution function on thephase space $H^{3}(M)$. This distribution function is a mixedHusimi/anti-Husimi distribution corresponding to the different normal orderingprescriptions for the string coupling and deviations of the complex structuremoduli. From the integral representation of this distribution function in termsof 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 thislimit.Comment: 27 pages. v2:reference and footnote adde