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

TL;DR

This paper develops a novel approach to Kerr Compton amplitudes using Nekrasov-Shatashvili functions, revealing their analytic properties, symmetries, and large frequency behavior, and compares these to traditional perturbative methods in General Relativity.

Contribution

It introduces a new method connecting Kerr Compton amplitudes with NS-functions of the confluent Heun equation, enabling analytic insights and higher-order amplitude calculations.

Findings

Analytic properties and symmetries of Compton amplitudes derived from NS-functions.

Agreement of large frequency behavior with numerical computations.

Explicit tree-level amplitude contributions up to order $a_{BH}^8$.

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.