One loop determinant in the extremal black hole from quasinormal modes

TL;DR

This paper computes the one-loop partition function of scalar fields in extremal black hole geometries using quasinormal modes, confirming divergence consistency with heat kernel methods and extending to near-extremal Kerr-Newman black holes.

Contribution

It introduces a method to evaluate one-loop determinants in extremal black holes via quasinormal modes, including higher spin fields and near-extremal cases.

Findings

Logarithmic divergence matches heat kernel results.

Partition functions reduce to extremal Reissner Nordström case in near-horizon limit.

Higher spin mode functions are not smooth at the horizon, requiring removal.

Abstract

In this paper, we evaluate the one-loop partition function of a scalar field in the near-horizon geometry of the extremal Reissner Nordstr\"{o}m black hole from an infinite product over quasinormal modes using the Denef-Hartnoll-Sachdev (DHS) formula. We show that the logarithmic divergent term of the one-loop partition function computed using the DHS formula agrees with the heat kernel method. Using the same formula, we also evaluate the one-loop partition function of a scalar field in the near-extremal Kerr-Newman black hole and observe that it reduces to the same in the near-horizon $AdS_2\times S^2$ geometry of the extremal Reissner Nordstr\"{o}m black hole when the angular velocity at the horizon is tuned to $2\pi T_{BH}$ value. We observe that, for higher spin fields, the mode functions are not smooth at the horizon for certain quasinormal frequencies; therefore, we remove them to obtain the one-loop determinant.