Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function

TL;DR

This work computes the logarithmic corrections to the entropy of extremal quarter-BPS black holes in ${\cal N}=4$ supergravity by evaluating the one-loop determinants of massless matter multiplet fluctuations in the near-horizon geometry $AdS_2 \times S^2$ using the quantum entropy function. Through detailed heat-kernel analyses of scalar, vector, $p$-form, and fermion fields, it shows that the bosonic and fermionic contributions within a matter multiplet cancel, yielding ${\Delta S^{\text{matter}}_{BH}=0}$ and hence a logarithmic correction that is independent of the number of matter multiplets. This finding aligns with microscopic results for these theories, which show no logarithmic dependence on the matter content. The study isolates the matter sector's role and emphasizes the need to evaluate gravity-multiplet contributions for the full macroscopic entropy, offering a framework for extensions to full string theory and potential insights via dual descriptions.

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.