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| Paper IPM / P / 7879 |
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| Abstract: | |||||
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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.
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