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Paper   IPM / P / 7879
School of Physics
Title:   Minimal redefinition of the OSV ensemble
Author(s):
 1 S. Parvizi 2 A.R. Tavanfar
Status:   Published
Journal: J. Math. Phys.
Vol.:  47
Year:  2006
Pages:   122304
Supported by:  IPM
Abstract:
An interesting conjecture, ZBH=|Ztop|2, as recently proposed by Ooguri, Strominger and Vafa, connects the free energy of topological strings with the degeneracy of 4-dimensional BPS black hole states arising in N=2 compactifications of IIA(B) string theories. In the original proposal the black hole ensemble is a mixed ensemble, in the sense that it treats the magnetic charges microcanonically while the electric charges canonically. Consequently, the results for degeneracy of states as obtained from the ensemble inverse-Laplace integration, suffer from prefactors which do not respect the (relevant) electric-magnetic dualities. One idea to overcome this deficiency, as claimed recently, is imposing a nontrivial measure for the ensemble sum. We address this problem and upon a redefinition of the OSV ensemble whose variables are as numerous as the electric potentials, show that for restoring the symmetry no non-Euclidean measure is needful. In detail, we rewrite the OSV free energy as a function of new variables which are combinations of the electric-potentials and the black hole charges. Subsequently the Legendre transformation which bridges between the entropy and the black hole free energy in terms of these variables, points to a generalized ensemble. In this context we will consider all the cases of relevance: small and large black holes, with or without D6-brane charge. For the case of vanishing D6-brane charge, the ensemble is pure canonical and the electric-magnetic duality is restored exactly, leading to the proper results for the black hole degeneracy of states. For more general cases, the construction still works well as far as the violation of the duality by the corresponding OSV result is restricted to a prefactor. In a concrete example we shall show that for black holes with non-vanishing D6-brane charge, there are cases where the duality violation goes beyond this restriction, thus imposing non-trivial measures is incapable of restoring the duality. This observation signals for a deeper modification in the OSV proposal.