Revisiting localization for BPS black hole entropy

A bstract Macroscopic computation of the degeneracy / index of a BPS black hole requires carrying out string theory path integral over the near horizon geometry of the black hole. It has been suggested that the path integral can be localized to a finite dimensional subspace using the unbroken supersymmetries of the near horizon geometry. We discuss several issues that need to be resolved before one can declare victory: (1) delta-function normalizable eigenmodes of the kinetic operator do not form a representation of the supersymmetry algebra since the Killing spinors grow exponentially as we approach the asymptotic boundary of the near horizon geometry, (2) the logarithmic contributions coming from various one loop determinants are sensitive to the ultraviolet cut-off and therefore without detailed knowledge of the ultraviolet cut-off one cannot determine the order one correction to the entropy that could depend on the ratios of the charges, (3) the results of index theorem indicate that there are additional flat directions of the localization action, at least at the quadratic order, than what has been considered so far. We also discuss a limited application of localization techniques to carry out the path integral over the zero modes of the metric and the gravitino, and get a finite answer. This is consistent with the results of arXiv:0905.2686 and shows that for computing BPS black hole entropy it is not necessary to regulate the path integral over these zero modes using the super-Schwartzian action.