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

**Authors:** Yilber Fabian Bautista, Giulio Bonelli, Cristoforo Iossa, Alessandro Tanzini, Zihan Zhou  
**arXiv:** [2312.05965](https://arxiv.org/abs/2312.05965)  
**Date:** 2023-12-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2312.05965v1/

## Key points

- Yilber Fabian Bautista and collaborators study Kerr Compton amplitudes within a partial wave basis
- They use the Nekrasov-Shatashvili (NS) function derived from the confluent Heun equation (CHE)
- Analytical properties and symmetries of CHE transfer to Compton amplitudes
- Analysis of Compton phase-shift dependence on Kerr spin parameter compared to post-Minkowskian perturbative approach in General Relativity
- Exploration of universal behavior at large frequencies of renormalized angular momentum associated with CHE
- Agreement between numerical computations and theoretical predictions
- Analytic continuation in terms of harmonics quantum number $\ell$ aligns with PM expansion outcomes for observables
- Derivation of contributions to tree-level gravitational Compton amplitude up to $\mathcal{O}(a_{\text{BH}}^8)$ without super-extremal limits for Kerr spin parameters
- Bridging black hole perturbation theory with conformal field theory (CFT$_2$) opens new avenues for theoretical exploration

## Summary

In this groundbreaking study, Yilber Fabian Bautista and collaborators delve into the realm of Kerr Compton amplitudes within a partial wave basis. They utilize the Nekrasov-Shatashvili (NS) function derived from the confluent Heun equation (CHE), which exhibits remarkable analytic properties and symmetries that seamlessly transfer to the Compton amplitudes under investigation. By leveraging this connection, the researchers meticulously analyze the analytical dependence of the Compton phase-shift on the Kerr spin parameter. This offers a direct comparison with the conventional post-Minkowskian (PM) perturbative approach within General Relativity (GR). Furthermore, their study delves into exploring the universal behavior at large frequencies of the characteristic exponent associated with CHE, also known as the renormalized angular momentum. Through rigorous numerical computations, agreement is found with theoretical predictions. The team also investigates the analytic continuation in terms of harmonics quantum number $\ell$ of partial waves. They demonstrate that transitioning to physical integer values aligns cohesively with PM expansion outcomes for observables. Moreover, by pushing boundaries in their analysis, they successfully derive contributions to tree-level gravitational Compton amplitude for point-particle interactions in a covariant basis up to $\mathcal{O}(a_{\text{BH}}^8)$ without necessitating super-extremal limits for Kerr spin parameters. This comprehensive study not only sheds light on intricate aspects of Kerr Compton amplitudes but also showcases a novel approach that bridges black hole perturbation theory with conformal field theory (CFT$_2$). This opens up new avenues for theoretical exploration at the intersection of these two domains.