Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions

TL;DR

This work develops a general framework to compute logarithmic corrections to the entropy of rotating extremal black holes via the quantum entropy function in Euclidean quantum gravity. It isolates contributions from non-zero and zero modes in the near-horizon AdS_2 geometry and applies the method to 4D extremal Kerr and 5D BMPV black holes (including CHL models and T^5), yielding precise agreement with microscopic string-theory results and clarifying ensemble choices and hair-mode removals. The analysis reveals non-universality in Cardy-limit expectations for these scaling limits and extends to slowly rotating cases, providing robust tests of microstate counting (Kerr/CFT expectations) and deepening our understanding of infrared quantum gravity corrections to black hole entropy. Overall, the paper demonstrates that infrared, low-energy data alone dictate the logarithmic entropy corrections and offers detailed matches between macroscopic and microscopic descriptions across a broad class of rotating extremal black holes.

Abstract

We compute logarithmic corrections to the entropy of rotating extremal black holes using quantum entropy function i.e. Euclidean quantum gravity approach. Our analysis includes five dimensional supersymmetric BMPV black holes in type IIB string theory on T^5 and K3 x S^1 as well as in the five dimensional CHL models, and also non-supersymmetric extremal Kerr black hole and slowly rotating extremal Kerr-Newmann black holes in four dimensions. For BMPV black holes our results are in perfect agreement with the microscopic results derived from string theory. In particular we reproduce correctly the dependence of the logarithmic corrections on the number of U(1) gauge fields in the theory, and on the angular momentum carried by the black hole in different scaling limits. We also explain the shortcomings of the Cardy limit in explaining the logarithmic corrections in the limit in which the (super)gravity description of these black holes becomes a valid approximation. For non-supersymmetric extremal black holes, e.g. for the extremal Kerr black hole in four dimensions, our result provides a stringent testing ground for any microscopic explanation of the black hole entropy, e.g. Kerr/CFT correspondence.