Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity

TL;DR

The paper tests quantum gravity corrections to extremal black hole entropy by computing one-loop determinants of massless fields in the near-horizon geometry $AdS_2\times S^2$ for ${\cal N}=4$ and ${\cal N}=8$ theories and the STU model. Using heat-kernel techniques and careful treatment of zero modes, the authors show that the macroscopic logarithmic corrections reproduce the microscopic index predictions: zero for quarter-BPS ${\cal N}=4$, a coefficient $-2$ for 1/8-BPS ${\cal N}=8$, and a $+1$ coefficient for half-BPS STU black holes, thereby providing a nontrivial test of one-loop quantum gravity and AdS$_2$/CFT$_1$ in these settings. The work also clarifies the roles of gravity multiplet fluctuations, discrete AdS$_2$ modes, and the zero-mode structure in determining the entropy corrections. Altogether, the results offer a robust quantitative link between macroscopic quantum gravity corrections and microscopic degeneracy formulas in supersymmetric theories.

Abstract

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N=4 and N=8 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N=4 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N=8 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS_2/CFT_1 correspondence. We also extend our analysis to N=2 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.