Minimal redefinition of the OSV ensemble

Abstract

In the interesting conjecture, ZBH=|Ztop|2, proposed by Ooguri, Strominger, and Vafa (OSV), the black hole ensemble is a mixed ensemble. So if working in the complex polarization, the black hole degeneracy of states as obtained from the ensemble inverse-Laplace integration, generically receives prefactors that do not respect the electric-magnetic duality. One way to handle this, as claimed recently, is working instead of the complex polarization in the real polarization. The other idea would be imposing nontrivial measures for the ensemble sum in the complex polarization. We address this problem in the complex polarization, which is canonical, and upon a redefinition of the OSV ensemble with variables as numerous as the electric potentials, show that for restoring the symmetry no non-Euclidean measure is needed. In detail, applying the electric-magnetic duality as a constraint governing the proper definition of the ensemble variables, we rewrite the OSV free energy as a function of new variables that 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 that is well behaved in the complex polarization. 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, the new ensemble is purely canonical and the electric-magnetic duality is restored exactly, leading to proper results for the black hole degeneracy of states to all orders in an asymptotic expansion. For more general cases as well, the construction does the job as far as the violation of the duality by the corresponding OSV result is restricted to a prefactor. In the case of black holes with nonvanishing D6-brane charge, in a concrete example, we shall show that there are cases where the duality violation goes beyond this restriction and imposing nontrivial measures is incapable of restoring the duality.