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One loop effective actions in Kerr-(A)dS Black Holes

Paolo Arnaudo, Giulio Bonelli, Alessandro Tanzini

TL;DR

The authors develop an exact analytic framework for computing one-loop scalar effective actions in Kerr-(A)dS black hole backgrounds in four and five dimensions by extending the Gelfand-Yaglom method to second-order operators with regular singularities and solving the resulting Heun equations via Nekrasov-Shatashvili functions. The determinants factorize into radial and angular parts, with angular eigenvalues determined by NS data and radial determinants expressed through Heun connection coefficients, enabling explicit accounting of quasi-normal mode contributions. The work provides comprehensive 4D Kerr-dS and 5D SAdS results, including reductions to SdS, dS, and pure AdS limits, and connects the Euclidean, thermal, and spectral data through zeta-function regularization and the DHS formula. These exact results offer new tools to study quantum fields in curved spacetimes, illuminate tidal and finite-size effects, and suggest extensions to higher spins and broader gravitational backgrounds with potential holographic applications.

Abstract

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the DHS formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.

One loop effective actions in Kerr-(A)dS Black Holes

TL;DR

The authors develop an exact analytic framework for computing one-loop scalar effective actions in Kerr-(A)dS black hole backgrounds in four and five dimensions by extending the Gelfand-Yaglom method to second-order operators with regular singularities and solving the resulting Heun equations via Nekrasov-Shatashvili functions. The determinants factorize into radial and angular parts, with angular eigenvalues determined by NS data and radial determinants expressed through Heun connection coefficients, enabling explicit accounting of quasi-normal mode contributions. The work provides comprehensive 4D Kerr-dS and 5D SAdS results, including reductions to SdS, dS, and pure AdS limits, and connects the Euclidean, thermal, and spectral data through zeta-function regularization and the DHS formula. These exact results offer new tools to study quantum fields in curved spacetimes, illuminate tidal and finite-size effects, and suggest extensions to higher spins and broader gravitational backgrounds with potential holographic applications.

Abstract

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the DHS formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.
Paper Structure (28 sections, 201 equations)

This paper contains 28 sections, 201 equations.

Table of Contents

  1. Introduction
  2. One-loop black hole effective actions and Gelfand-Yaglom theorem
  3. Kerr-de Sitter spacetime in four dimensions
  4. Angular Problem
  5. Radial Problem and expression for the determinant
  6. Determinant of radial operator
  7. Schwarzschild-de Sitter spacetime in four dimensions
  8. Pure de Sitter spacetime in four dimensions
  9. Reduction of the determinant from Schwarzschild-de Sitter to pure de Sitter
  10. Schwarzschild anti-de Sitter spacetime in five dimensions
  11. Determinant of radial operator
  12. Pure anti-de Sitter spacetime in five dimensions
  13. Detailed analysis of the effective actions
  14. Wick rotation and the thermal version
  15. Gelfand-Yaglom theorem
  16. ...and 13 more sections

Theorems & Definitions (1)

  • Remark A.1