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.

arXiv comment34 pages, 2 figures

References in corpus (63)

• Lectures on holographic methods for condensed matter physics
• Soft Hair on Black Holes
• Aspects of the BMS/CFT correspondence
• W-symmetry in Conformal Field Theory
• The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant
• Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions
• The BMS/GCA correspondence
• Superrotation Charge and Supertranslation Hair on Black Holes
• Integrable Theory of the Perturbation Equations
• Galilean Conformal Algebras and AdS/CFT
• How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples
• Supertranslations and Superrotations at the Black Hole Horizon
• Holography of 3d Flat Cosmological Horizons
• Asymptotic Structure of Symmetry Reduced General Relativity
• Entropy of three-dimensional asymptotically flat cosmological solutions
• String Lessons for Higher-Spin Interactions
• Entanglement entropy in Galilean conformal field theories and flat holography
• Lifshitz holography
• GCA in 2d
• On the geometry of higher-spin gauge fields
• Flat-Space Chiral Gravity
• Extended Symmetries at the Black Hole Horizon
• Notes on the BMS group in three dimensions: I. Induced representations
• The Bi-Hamiltonian Structure of the Perturbation Equations of KdV Hierarchy
• Higher spin theory in 3-dimensional flat space
• Asymptotically flat spacetimes in three-dimensional higher spin gravity
• Tensionless Strings and Galilean Conformal Algebra
• Generalized Black Holes in Three-dimensional Spacetime
• Tensionless Strings from Worldsheet Symmetries
• On the null origin of the ambitwistor string
• Asymptotic symmetries and dynamics of three-dimensional flat supergravity
• Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity
• Boundary conditions for General Relativity on AdS${}_3$ and the KdV hierarchy
• Chemical potentials in three-dimensional higher spin anti-de Sitter gravity
• Near-Horizon Geometry and Warped Conformal Symmetry
• Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes
• Higher-Spin Geometry and String Theory
• The BMS Bootstrap
• Holographic Metamagnetism, Quantum Criticality, and Crossover Behavior
• A Grassmann Path From AdS_3 to Flat Space
• Higher Spin Black Holes with Soft Hair
• Coupled KdV equations of Hirota-Satsuma type
• Three-dimensional supergravity reloaded
• Super-BMS${}_3$ algebras from $\mathcal{N}=2$ flat supergravities
• Tensionless Superstrings: View from the Worldsheet
• Asymptotic structure of $\mathcal{N}=2$ supergravity in 3D: extended super-BMS${}_3$ and nonlinear energy bounds
• The nuts and bolts of the BMS Bootstrap
• Supersymmetric Extension of GCA in 2d
• W Symmetry and Integrability of Higher spin black holes
• Flat space (higher spin) gravity with chemical potentials
• Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics
• Emergent scale invariance of disordered horizons
• Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond
• Asymptotically flat structure of hypergravity in three spacetime dimensions
• Spectral Flow in 3D Flat Spacetimes
• Free-field realisations of the BMS${}_3$ algebra and its extensions
• ${\mathrm{BMS}}_3$ invariant fluid dynamics at null infinity
• Boundary dynamics of asymptotically flat 3D gravity coupled to higher spin fields
• The complex null string, Galilean conformal algebra and scattering equations
• Gravitational Couplings of Higher Spins from String Theory
• Super-GCA from $\mathcal{N}=(2,2)$ Super-Virasoro
• Log corrections to entropy of three dimensional black holes with soft hair
• Addendum to "Super-GCA from $\mathcal{N}=(2,2)$ super-Virasoro": Super-GCA connection with tensionless strings

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• 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