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Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$

Marcela Cárdenas, Francisco Correa, Miguel Pino

TL;DR

This work develops a constructive bridge between integrable systems and AdS$_3$ gravity by imposing relaxed boundary conditions that realize two copies of the AKNS hierarchy as the asymptotic dynamics. Through a Chern-Simons/geometry correspondence, it derives field-dependent Killing vectors, constructs finite (yet integrable) canonical charges, and shows these charges form an abelian algebra guided by AKNS bi-Hamiltonian structure. The analysis yields concrete black hole solutions with nontrivial AKNS charges, computes mass and entropy, and proves that stationary AKNS black holes have a constant temperature governed by a hyperelliptic spectral curve. A KdV/cnoidal example illustrates the spectrum and thermodynamics, highlighting deep links between the geometry of hyperelliptic curves and gravitational thermodynamics. The results offer a new holographic perspective where integrable hierarchies organize boundary dynamics and black hole thermodynamics, with potential extensions to other gauge theories and generalized Gibbs ensembles.

Abstract

This work deepens the study of integrable asymptotic symmetries for AdS$_{3}$. They are given by an infinite set of integrable nonlinear equations known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, characterized by an also infinite set of abelian conserved charges. We present their field-dependent Killing vectors and the computation of the canonical charges associated to the asymptotic metric, together with their corresponding charge algebra. We study black hole thermodynamics and show that the temperature for stationary black holes falling in the AKNS asymptotics is always constant, even in the case where the solutions are not axisymmetric. This is related to the existence of a hyperelliptic curve, which appears as a fundamental object in many integrable systems. We also present a special solution associated with the Korteweg-de Vries equation, that is a particular case of the AKNS integrable hierarchy. It is presented in the form of a periodic soliton leading to a cnoidal KdV black hole, whose temperature is characterized by two copies of hyperelliptic curves.

Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$

TL;DR

This work develops a constructive bridge between integrable systems and AdS gravity by imposing relaxed boundary conditions that realize two copies of the AKNS hierarchy as the asymptotic dynamics. Through a Chern-Simons/geometry correspondence, it derives field-dependent Killing vectors, constructs finite (yet integrable) canonical charges, and shows these charges form an abelian algebra guided by AKNS bi-Hamiltonian structure. The analysis yields concrete black hole solutions with nontrivial AKNS charges, computes mass and entropy, and proves that stationary AKNS black holes have a constant temperature governed by a hyperelliptic spectral curve. A KdV/cnoidal example illustrates the spectrum and thermodynamics, highlighting deep links between the geometry of hyperelliptic curves and gravitational thermodynamics. The results offer a new holographic perspective where integrable hierarchies organize boundary dynamics and black hole thermodynamics, with potential extensions to other gauge theories and generalized Gibbs ensembles.

Abstract

This work deepens the study of integrable asymptotic symmetries for AdS. They are given by an infinite set of integrable nonlinear equations known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, characterized by an also infinite set of abelian conserved charges. We present their field-dependent Killing vectors and the computation of the canonical charges associated to the asymptotic metric, together with their corresponding charge algebra. We study black hole thermodynamics and show that the temperature for stationary black holes falling in the AKNS asymptotics is always constant, even in the case where the solutions are not axisymmetric. This is related to the existence of a hyperelliptic curve, which appears as a fundamental object in many integrable systems. We also present a special solution associated with the Korteweg-de Vries equation, that is a particular case of the AKNS integrable hierarchy. It is presented in the form of a periodic soliton leading to a cnoidal KdV black hole, whose temperature is characterized by two copies of hyperelliptic curves.
Paper Structure (17 sections, 116 equations)

This paper contains 17 sections, 116 equations.

Table of Contents

  1. Introduction
  2. Geometrization of AKNS
  3. A review on the AKNS integrable nonlinear hierarchy
  4. Lax pairs and stationary hierarchies
  5. Chern-Simons gravity, the zero curvature equation and the AKNS integrable system
  6. Asymptotic structure
  7. AKNS symmetries
  8. Canonical charges
  9. Symmetry transformations, integrability conditions
  10. Dynamical systems, Poisson structure and asymptotic symmetry algebras
  11. Abelian algebra
  12. Black hole Thermodynamics
  13. Regularity conditions and temperature
  14. The KdV black holes and cnoidal gravitational solutions
  15. Discussion
  16. ...and 2 more sections