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 [1]; 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 Sd−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.
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Acknowledgments
It is a great pleasure to thank Dionysios Anninos, Frederik Denef, Sean Hartnoll, Daniel Jafferis, and Gabriel Wong for stimulating conversations, and especially Adam Ball and Alejandra Castro for useful discussions and comments on the draft. AL was supported in part by the Croucher Foundation and the Black Hole Initiative at Harvard University. MG and KP were supported in part by the U.S. Department of Energy grant de-sc0011941.
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Grewal, M., Law, Y.T.A. & Parmentier, K. Black hole horizon edge partition functions. J. High Energ. Phys. 2023, 25 (2023). https://doi.org/10.1007/JHEP06(2023)025
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DOI: https://doi.org/10.1007/JHEP06(2023)025