Heat Kernel Expansion and Extremal Kerr-Newmann Black Hole Entropy in Einstein-Maxwell Theory

TL;DR

The paper targets quantum corrections to the entropy of extremal Kerr–Newman black holes within Einstein–Maxwell theory by extracting the Seeley–DeWitt coefficient $a_4$ for metric and gauge field fluctuations. It derives $a_4(x) = {1\over 360 \times 16\pi^2}\left(398 R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} + 52 R_{\mu\nu} R^{\mu\nu}\right)$, after gauge fixing, ghost contributions, and using background equations of motion, with terms involving $\bar{F}_{\mu\nu}$ canceling and duality invariance manifest. This infrared data is then applied to the extremal Kerr–Newman near-horizon geometry, including zero-mode counting, to obtain the logarithmic correction to the entropy; Kerr and extremal RN limits reproduce known results, providing a precise constraint on any microscopic or CFT-based explanations. The work clarifies how horizon geometry and zero modes govern the subleading $\ln A_H$ term and offers a concrete target for any dual description of black hole microphysics.

Abstract

We compute the second Seely-DeWitt coefficient of the kinetic operator of the metric and gauge fields in Einstein-Maxwell theory in an arbitrary background field configuration. We then use this result to compute the logarithmic correction to the entropy of an extremal Kerr-Newmann black hole.