Logarithmic corrections to $$ \mathcal{N} = {4} $$ and $$ \mathcal{N} = {8} $$ black hole entropy: a one loop test of quantum gravity

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 /CF T 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.

Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy

“…A particularly non-trivial test of this proposal is that expanding in quadratic fluctuations about this saddle point yields a term proportional to log a, where a is the radius of the AdS 2 and S 2 submanifolds. This matches precisely with the log a term extracted from the microscopic degeneracy as in section 2.1, in that both of them are zero 5 [24,25]. At this point, it is natural to ask if the exponentially suppressed corrections to the microscopic degeneracy have a proposed counterpart in the quantum entropy function formalism, and it turns out that the answer is yes.…”

“…4 We write the solution in the supergravity obtained by compactifying Type IIB Supergravity on K3. 5 The corresponding computation for black holes in N = 8 string theory yields a non-vanishing quantity which is also found to match with the microscopic formula [25].…”

“…In particular, we shall discuss the microscopic formula for black hole entropy obtained from string theory, and the Quantum Entropy Function formalism [16] which is a prescription for computing the full quantum entropy associated with the black hole horizon. We then propose a quantum test of the proposal of [16] analogous to the tests performed in [24,25].…”

“…This proposal has been tested in a variety of ways, for which we refer the reader to [15,[17][18][19][20][21][22] and more generally the lectures [23] for an overview. A particularly non-trivial test of this proposal is that the leading quantum corrections in the large charge limit, which scale as log (charges), to the semi-classical Bekenstein-Hawking formula as predicted from the string computation can be reproduced from the quantum entropy function for N = 4 and N = 8 string theory [24,25]. This is obtained by expanding the quantum entropy function about the saddle-point defined by the near-horizon geometry of the black hole.…”

Non-renormalization for non-supersymmetric black holes

“…It is known that these large logarithms offer an infrared window into ultraviolet physics: they are computable in the low energy theory and yield precision data that must be matched by sub-leading terms in the asymptotic density of black hole microstates [5][6][7]. Agreement with the microscopic theory has been established in those (highly supersymmetric) cases where precision counting is available [8][9][10]. We discuss these logarithms for non-supersymmetric black holes using effective quantum field theory.…”