Revisiting the logarithmic corrections to the black hole entropy

Logarithmic corrections to the entropy of extremal black holes have been successfully used to accurately match degeneracies from microscopic constructions to calculations of the gravitational path integral. In this paper, we revisit the problem of deriving such corrections for the case of extremal black holes, either non-supersymmetric or supersymmetric, and for near-extremal black holes. The zero-modes that are present at extremality are crucial, since their path integral cannot be treated quadratically and needs to be regulated. We show how the regulated result can be obtained by taking the zero-temperature limit of either the 4d Einstein-Maxwell or 4d supergravity path integral to find the Schwarzian or super-Schwarzian theories. This leads to drastically different estimates for the degeneracy of extremal black hole: it vanishes when the extremal limit does not preserve supersymmetry, while it reproduces Bekenstein-Hawking in the BPS case. In a companion paper, we discuss how such zero-modes affect the calculation of BPS black holes degeneracies, using supersymmetric localization for an exact computation of the gravitational path integral.