Logarithmic corrections to extremal black hole entropy in N = 2 $$ \mathcal{N}=2 $$ , 4 and 8 supergravity

Abstract

We compute the logarithmic correction to black hole entropy about exponentially suppressed saddle points of the Quantum Entropy Function corresponding to ℤ N orbifolds of the near horizon geometry of the extremal black hole under study. By carefully accounting for zero mode contributions we show that the logarithmic contributions for quarter-BPS black holes in $$ \mathcal{N}=4 $$ supergravity and one-eighth BPS black holes in $$ \mathcal{N}=8 $$ supergravity perfectly match with the prediction from the microstate counting. We also find that the logarithmic contribution for half-BPS black holes in $$ \mathcal{N}=2 $$ supergravity depends non-trivially on the ℤ N orbifold. Our analysis draws heavily on the results we had previously obtained for heat kernel coefficients on ℤ N orbifolds of spheres and hyperboloids in arXiv:1311.6286 and we also propose a generalization of the Plancherel formula to ℤ N orbifolds of hyperboloids to an expression involving the Harish-Chandra character of sl (2, R), a result which is of possible mathematical interest.