Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function (1005.3044v2)

Abstract: We evaluate the one loop determinant of matter multiplet fields of N=4 supergravity in the near horizon geometry of quarter BPS black holes, and use it to calculate logarithmic corrections to the entropy of these black holes using the quantum entropy function formalism. We show that even though individual fields give non-vanishing logarithmic contribution to the entropy, the net contribution from all the fields in the matter multiplet vanishes. Thus logarithmic corrections to the entropy of quarter BPS black holes, if present, must be independent of the number of matter multiplet fields in the theory. This is consistent with the microscopic results. During our analysis we also determine the complete spectrum of small fluctuations of matter multiplet fields in the near horizon geometry.

Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function

The paper, authored by Shamik Banerjee, Rajesh K. Gupta, and Ashoke Sen, investigates logarithmic corrections to the entropy of extremal black holes, particularly those with a quarter BPS nature in the framework of $N=4$ supergravity. The authors employ the quantum entropy function methodology to achieve a comprehensive understanding of these corrections, extending beyond the semiclassical Bekenstein-Hawking area law.

The research focuses on calculating the one-loop determinants of matter multiplet fields in the near-horizon geometry, exploiting the supersymmetric background to streamline calculations. The central result presented is the vanishing of the net contribution from all fields in the matter multiplet to the logarithmic corrections of black hole entropy. This is a key result, consistent with the precise microscopic counting of states obtained in string theory for quarter BPS black holes within the same supersymmetric framework.

The paper details the methodology through six primary sections, beginning with a thorough theoretical groundwork that includes Wald’s entropy formula generalization and its applicability limitations due to higher derivative terms and classical theories' non-local nature. The authors then introduce the quantum entropy function formalism, connecting it with the AdS$_2$/CFT$_1$ correspondence, allowing the capture of microscopic string theory states' non-perturbative features.

The authors conduct a meticulous evaluation of the fluctuations in massless fields from the vector, p-form, and fermionic components, ensuring a quantum correction analysis aligned with the $N=4$ supergravity gauge symmetries and dynamics. Their computations reveal fascinating simplifications due to the high degree of supersymmetry in the geometry, such as factorizing problems into scalar and vector field eigenstates or dealing with AdS$_2$ and S$_2$ spaces.

Notably, the analysis of the quantum path integral over fluctuations excludes non-perturbative contributions from stringy modes and Kaluza-Klein modes. The work further distinguishes massive excitations' contributions, which vanish at one-loop, confirming compatibility with supersymmetry expectations and existing literature relating to logarithmic corrections within the Cardy limit.

Future implications of this research are broad and interdisciplinary, providing a fertile ground for comparisons between macroscopic and microscopic perspectives on black hole entropy. The findings urge further speculative pursuits, specifically regarding the one-loop gravity multiplet's contribution, which remains computationally challenging due to its added complexity from the gauge and gravitational backgrounds. Moreover, potential explorations into the $N=8$ supersymmetric theories where non-zero corrections exist are highlighted as a promising domain.

Overall, this paper stands as a substantial advance in understanding entropy corrections for BPS black holes, leveraging conformal field theories' insights to resolve gravitational puzzles. Future advances could potentially harness localized computations or fully realized string theoretic evaluations, leading to a deeper union between gravitational and quantum theories in high-dimensional and supersymmetric contexts.