The Algebraic Structure Underlying Pole-Skipping Points

TL;DR

This work shows that a discrete tower of pole-skipping points in holographic Green's functions suffices to reconstruct the full bulk metric of static, planar black holes analytically. The authors develop a boundary-to-bulk map that expresses near-horizon coefficients $g_{vv_n}$ and $g_{vr_{n-1}}$ in terms of pole-skipping data, the frequency input $\omega_1$, and elementary symmetric polynomials $E_n(\mu^m)$, by solving a system of linear equations and employing Viète relations. They extend the method to massive probes, beyond the probe limit, Lifshitz/hyperscaling-violating geometries, and $T\bar{T}$-deformed boundaries, and reinterpret the vacuum Einstein equations as pole-skipping constraints. A central finding is the emergence of universal $\mu$-polynomial constraints $P_n(\mu^m)=0$ that relate pole-skipping points across all backgrounds and perturbation channels, reinforcing that bulk geometry is redundantly encoded in pole-skipping data. The results suggest a powerful, purely analytical route to bulk reconstruction and expose deep algebraic structures underlying holographic pole-skipping phenomena with potential implications for quantum chaos, holographic bounds, and spacetime emergence.

Abstract

The holographic Green's function becomes ambiguous, taking the indeterminate form `$0/0$', at an infinite set of special frequencies and momenta known as ``pole-skipping points''. In this work, we propose that these pole-skipping points can be used to reconstruct both the interior and exterior geometry of a static, planar-symmetric black hole in the bulk. The entire reconstruction procedure is fully analytical and only involves solving a system of linear equations. We demonstrate its effectiveness across various backgrounds, including the BTZ black hole, its $T\bar{T}$-deformed counterparts, as well as geometries with Lifshitz scaling and hyperscaling-violation. Within this framework, other geometric quantities, such as the vacuum Einstein equations, can also be reinterpreted directly in terms of pole-skipping data. Moreover, our approach reveals a hidden algebraic structure governing the pole-skipping points of Klein-Gordon equations of the form $(\nabla^{2} + V(r))φ(r) = 0$: only a subset of these points is independent, while the remainder is constrained by an equal number of homogeneous polynomial identities in the pole-skipping momenta. These identities are universal, as confirmed by their validity across a broad class of bulk geometries with varying dimensionality, boundary asymptotics, and perturbation modes.

Figures (4)

• Figure 1: From left to right, the figure displays heatmaps showing the values of $\log|\mathcal{G}^{\mathcal{O}}_{R}(\mathrm{Im}\omega,\mathrm{Im}k)|$ for different values of $\Delta$: $\frac{4}{3}$, $\frac{7}{3}$, and $\frac{10}{3}$ (after taking $T_{b}=\frac{1}{2\pi}$). The red lines and blue lines represent the poles and zeroes of Eq. \ref{['equ_BTZ_GR']} respectively. The black dots depict the pole-skipping points obtained from the near-horizon analysis. In all panels, the locations of the black dots precisely coincide with the intersections of poles and zeroes of $\mathcal{G}^{\mathcal{O}}_{R}(\omega,k)$.
• Figure 2: The figure displays heatmaps showing the values of $\log|\mathcal{G}^{\mathcal{O}}_{z_c}(\mathrm{Im}\omega,\mathrm{Im}k,z_{c})|$ for different values of $z_{c}$: $\frac{5}{1000}$, $\frac{3}{10}$, and $\frac{5}{10}$ (with $T=\frac{1}{2\pi}$ and $\Delta=\frac{4}{3}$). The red and blue lines represent the poles and zeroes of Eq. \ref{['equ_TTbar_GR']}, respectively. The black dots indicate the pole-skipping points obtained from the near-horizon analysis of the BTZ black hole described by Eq. \ref{['equ_intersections_poles_zeroes_BTZ']}. In all panels, the black dots align precisely with the intersections of poles and zeroes of $\mathcal{G}^{\mathcal{O}}_{z_c}(\omega,k,z_{c})$, implying that the pole-skipping points remain invariant under $T\bar{T}$ deformation.
• Figure 3: From left to right, the higher-resolution version of the third panel of Figure \ref{['fig_TTbar_GR']} around different values of $\omega$: $-5i$, $-6i$, and $-7i$.
• Figure 4: For each row, from left to right, the figure compares the "common" pole-skipping points $\mu_{n,q}$ obtained by solving $\text{Det}(\mathcal{M}^{(n)}(\boldsymbol{\mu})) = 0$ (depicted by the red dashed line) with those determined by incorporating $P_{n}(\mu^{m})= 0$ (depicted by the black solid line) for $n = 3, 4, 5$, spanning the full parameter space of $K$ and $\tau$. From top to bottom, the rows correspond to $K = 1$, $0$, and $-1$, respectively.