The Algebraic Structure Underlying Pole-Skipping Points

TL;DR

This paper shows how pole-skipping points in holographic Green's functions can be used to analytically reconstruct black hole geometries and reveal universal algebraic structures governing these special frequencies and momenta.

Contribution

It introduces a fully analytical method to reconstruct bulk geometries from pole-skipping data and uncovers universal polynomial identities linking pole-skipping points across diverse backgrounds.

Findings

Successful reconstruction of various black hole geometries from pole-skipping points.

Reinterpretation of Einstein equations in terms of pole-skipping data.

Discovery of universal algebraic identities constraining pole-skipping points.

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 $(\nabla^{2} + V(r))\phi(r) = 0$: 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.