Black hole perturbation theory meets ${\mathrm{CFT}}_2$: Kerr-Compton amplitudes from Nekrasov-Shatashvili functions

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 $\mathcal{O}(a_{\mathrm{BH}}^8)$, without the need to take the superextremal limit for Kerr spin.

Figure 1

Numerical evaluation of the high-frequency behavior of renormalized angular momentum $\nu$ using the Black Hole Perturbation Toolkit [63]. The solid lines represent $\nu$ for various perturbation parameters. The dashed line is the analytic estimation from (18).

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).