Duality and four-dimensional black holes: gravitational waves, algebraically special solutions, pole skipping, and the spectral duality relation in holographic thermal CFTs

TL;DR

This work investigates duality structures in four-dimensional black hole spacetimes, focusing on linearised perturbations of gravity and electromagnetism and their holographic implications for three-dimensional CFTs. The authors formulate dualities via Darboux transformations between even (longitudinal) and odd (transverse) perturbation channels, identifying algebraically special frequencies $ω_*$ that govern these relations. In AdS, they derive the spectral duality relation linking the longitudinal and transverse quasinormal mode spectra through meromorphic boundary correlators, enabling one-channel spectra to determine the other. They illustrate the framework with Schwarzschild, Reissner–Nordström, and linear axion models, discuss pole skipping, and connect bulk constraints to boundary thermal observables.

Abstract

The physics of gravitational waves and other classical fields on specifically four-dimensional backgrounds of black holes exhibits electric-magnetic-like dualities. In this paper, we discuss the structure of such dualities in terms of geometrical quantities with a physically-intuitive interpretation. In turn, we explain the interplay between the algebraic structure of black hole spacetimes and their associated dualities. For large classes of black hole geometries, explicit constructions are presented. We then use these results and apply them to the holographic study of three-dimensional conformal field theories (CFTs), discussing how such dualities place stringent constraints on the thermal spectra of correlators. In particular, the dualities enforce the recently-developed spectral duality relation along with a multitude of implications for the physics of thermal CFTs. A number of numerical results supporting our conclusions is also presented, including a demonstration of how the longitudinal spectrum of quasinormal modes determines the transverse spectrum, and vice versa.

Figures (8)

• Figure 1: The poles of $G_+$ (black squares) and $G_-$ (red diamonds) at $k=0.1$. In both channels, the poles converge to the Matsubara frequencies at zero momentum. The longitudinal channel exhibits a hydrodynamic charge diffusion mode.
• Figure 2: The zeroes of $S(\omega)$ associated with $G_\pm^J$. As we lower momentum, the asymptotic branches collide with one another, and poles converge to the Matsubara frequencies in the zero-momentum limit.
• Figure 3: Left panel: The diffusive mode interpolates the pole-skipping points from Eq. \ref{['eq:PS2']} at integer multiples of the negative imaginary Matsubara frequencies. Right panel: The difference $\omega_\text{diff}-\omega_*$ vanishes at the pole-skipping points, and rapidly decays to zero as real $k$ is increased. This implies that the function $-\omega_*(k)$ can be used as an excellent large-$k$ approximation of the diffusive dispersion relation.
• Figure 4: The zeroes of $S(\omega)$ for the energy-momentum tensor correlators at $k=5$. The lower complex half-plane contains the longitudinal spectrum (squares), whereas the upper complex half-plane contains the transverse spectrum (diamonds). The set of all zeroes is split into the hydrodynamic modes (filled), gapped modes (empty), and the algebraically special frequency (red dot), which acts as a 'faux longitudinal mode'. The dependence of the sound modes on real $k \in [0,14]$ is shown with a black line.
• Figure 5: Numerical verification of the spectral duality relation \ref{['eq:SDR']} for $k=5$ and $n_\text{max}=0,3,5,10,20$. Here, $\omega$ is imaginary for illustrative purposes. It is clear that the odd part of $S_{n_\text{max}}(\omega)$ converges to the sine function, satisfying the spectral duality relation.