Integrable systems with BMS${}_3$ Poisson structure and the dynamics of locally flat spacetimes

Abstract

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS${}_3$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis is performed in terms of two-dimensional gauge fields for $isl(2,R)$. Although the algebra is not semisimple, the formulation can be carried out à la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer $k$, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For $k\ge 1$, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, and they are in involution; while in the case of $k=0$, they generate the BMS${}_3$ algebra. In the special case of $k=1$, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to a specific type of the Hirota-Satsuma coupled KdV systems. For $k\ge 1$, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of $k$. Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is recovered from the corresponding surface integrals in the canonical approach.

34 pages, 2 figures

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Cited by in corpus (22)

• Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite $\sqrt{T\bar{T}}$ deformations
• Carroll black holes
• Dynamical structure of Carrollian Electrodynamics
• Nonlinear automorphism of the conformal algebra in 2D and continuous $\sqrt{T\bar{T}}$ deformations
• Near horizon dynamics of three dimensional black holes
• Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies
• Asymptotic dynamics of AdS${}_3$ gravity with two asymptotic regions
• Revisiting the asymptotic dynamics of General Relativity on AdS${}_3$
• KdV-charged black holes
• Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy
• Hypersymmetric extensions of Maxwell Chern-Simons gravity in (2+1) dimensions
• Integrable systems and the boundary dynamics of higher spin gravity on AdS${}_3$
• Gravity coupled to a scalar field from a Chern-Simons action: describing rotating hairy black holes and solitons with gauge fields
• BTZ black hole with KdV-type boundary conditions: Thermodynamics revisited
• Asymptotic Symmetries of Three-Dimensional Black Strings
• The anisotropic chiral boson
• Phase space holography with no strings attached
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Remarks on BMS${}_3$ invariant field theories: correlation functions and nonunitary CFTs
• 1/c deformations of AdS${}_3$ boundary conditions and the Dym hierarchy
• AdS${}_3$ Carroll gravity: asymptotic symmetries and C-thermal configurations
• Polyharmonic Green Functions and Nonlocal BMS Transformations of a Free Scalar Field