  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.
