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

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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 AdS₂/CFT₁ 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.