Black Hole Horizon Edge Partition Functions

TL;DR

This paper generalizes a 1-loop black hole determinant formula to spinning fields in higher dimensions, revealing a bulk-edge split in the Euclidean partition function linked to quasinormal modes and surface degrees of freedom.

Contribution

It introduces an unambiguous bulk-edge split in the 1-loop Euclidean partition function for tensor fields of arbitrary spin on static black holes, connecting edge modes to quasinormal modes.

Findings

Edge partition function relates to lowest overtone quasinormal modes.

Bulk part corresponds to the renormalized thermal partition function.

Edge modes are associated with degrees of freedom on the bifurcation surface.

Abstract

We extend a formula for 1-loop black hole determinants by Denef, Hartnoll, and Sachdev (DHS) to spinning fields on any $(d+1)$-dimensional static spherically symmetric black hole. By carefully analyzing the regularity condition imposed on the Euclidean eigenfunctions, we reveal an unambiguous bulk-edge split in the 1-loop Euclidean partition function for tensor fields of arbitrary integer spin: the bulk part captures the "renormalized" thermal canonical partition function recently discussed in arXiv:2207.07024; the edge part is related to quasinormal modes (QNMs) that fail to analytically continue to a subset of Euclidean modes with enhanced fall-offs near the origin. Since the edge part takes the form of a path integral on $S^{d-1}$, this suggests that these are associated with degrees of freedom living on the bifurcation surface in the Lorentzian two-sided black hole geometry. For massive higher spin on static BTZ and massive vector on Nariai black holes, we find that the edge partition function is related to the QNMs with lowest overtone numbers.