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Black Hole Perturbation Theory Meets CFT$_2$: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions

Yilber Fabian Bautista, Giulio Bonelli, Cristoforo Iossa, Alessandro Tanzini, Zihan Zhou

TL;DR

The paper builds a bridge between black hole perturbation theory and two-dimensional conformal field theory by expressing Kerr Compton amplitudes in a partial-wave basis through the Nekrasov-Shatashvili NS function that solves the confluent Heun equation. This yields nonperturbative, spin-dependent information and a direct PM-CFT correspondence that recasts MST resummations in terms of NS data, including a clear near-far zone factorization. It demonstrates that PM and generic-ell expansions commute, with spurious poles canceling between near and far contributions, and shows how a covariant, tree-level Kerr Compton amplitude can be obtained to $O(a_{\text{BH}}^8)$ without invoking a super-extremal spin limit. The work also clarifies the analytic structure of phase shifts, highlighting two pole families tied to bound states and QNMs and opening avenues for non-linear and self-force extensions within a unified NS/CFT framework.

Abstract

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase-shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within General Relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE -- also known as the renormalized angular momentum -- and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number $\ell$ of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree level, point-particle, gravitational Compton amplitude in a covariant basis through $O(a_{\text{BH}}^8)$, without the need to take the super-extremal limit for Kerr spin.

Black Hole Perturbation Theory Meets CFT$_2$: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions

TL;DR

The paper builds a bridge between black hole perturbation theory and two-dimensional conformal field theory by expressing Kerr Compton amplitudes in a partial-wave basis through the Nekrasov-Shatashvili NS function that solves the confluent Heun equation. This yields nonperturbative, spin-dependent information and a direct PM-CFT correspondence that recasts MST resummations in terms of NS data, including a clear near-far zone factorization. It demonstrates that PM and generic-ell expansions commute, with spurious poles canceling between near and far contributions, and shows how a covariant, tree-level Kerr Compton amplitude can be obtained to without invoking a super-extremal spin limit. The work also clarifies the analytic structure of phase shifts, highlighting two pole families tied to bound states and QNMs and opening avenues for non-linear and self-force extensions within a unified NS/CFT framework.

Abstract

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase-shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within General Relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE -- also known as the renormalized angular momentum -- and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree level, point-particle, gravitational Compton amplitude in a covariant basis through , without the need to take the super-extremal limit for Kerr spin.
Paper Structure (15 sections, 86 equations, 3 figures)

This paper contains 15 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. spin-$s$ perturbations off Kerr
  3. NS-Function, MST PM-Resummation, and the High Frequency limit
  4. Generic-$\ell$ vs fixed-$\ell$ prescriptions
  5. Far-zone, tree-level gravitational Compton Amplitude for Kerr
  6. Discussion
  7. Appendix A: MST Method Review and Near-Far Factorization
  8. MST Method Review
  9. Near-Far Factorization
  10. phase-shifts for $s=0,\ell=1,m=1$ Perturbations
  11. Appendix B: CFT Method Review
  12. Connection Formula for CHE
  13. Solving Radial TME
  14. Computation of $F$ and Symmetry Properties
  15. Appendix C: Proof of Large Frequency Behavior

Figures (3)

  • Figure 1: Numerical evaluation of the high frequency behavior of renormalized angular momentum $\nu$ using Black Hole Perturbation Toolkit BHPToolkit. The solid lines represent $\nu$ for various perturbation parameters. The dashed line is the analytic estimation from \ref{['eq: analytic estimation nu']}.
  • Figure 2: A schematic diagram illustrates the convergence radius of near and far zone solutions. The blue region denotes the overlapping region where the matching is performed.
  • Figure 3: Arm length $A_{\tilde{Y}} (s)=4$ (white circles) and leg length $L_Y(s)=2$ (black dots) of a box at the site $s = (2,2)$ for the pair of superimposed diagrams $Y$ (solid lines) and $\tilde{Y}$ (dotted lines).