Higher spin Lifshitz theories and the Korteweg-de Vries hierarchy

Beccaria, Matteo; Gutperle, Michael; Li, Yi; Macorini, Guido

doi:10.1103/physrevd.92.085005

Cited by 17 publications

(15 citation statements)

References 53 publications

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“…Further interesting links between certain well-known classes of integrable systems and higher spin gravity on AdS 3 have been explored in [104], [105], [106], [68], [107]. read H (0)…”

Section: Extensions Of Our Resultsmentioning

confidence: 99%

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

Fuentealba

Matulich

Pérez

et al. 2018

J. High Energ. Phys.

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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), being isomorphic to the Poincaré algebra in 3D. Although the algebra is not semisimple, the formulation can still 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 ≥ 1, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, so that 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 the ones of a specific type of the Hirota-Satsuma coupled KdV systems. For k ≥ 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 then recovered from the corresponding surface integrals in the canonical approach.

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“…Further interesting links between certain well-known classes of integrable systems and higher spin gravity on AdS 3 have been explored in [104], [105], [106], [68], [107]. read H (0)…”

Section: Extensions Of Our Resultsmentioning

confidence: 99%

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

Fuentealba

Matulich

Pérez

et al. 2018

J. High Energ. Phys.

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show abstract

“…For the Lifshitz case, asymptotically Lifshitz boundary conditions based on the solution (1.3) have been described in [25,19,26,27,28]. In section 5, we comment on the extension of our analysis to asymptotically Lifshitz solutions, and argue that the boundary conditions of [19] could be re-interpreted as a novel kind of asymptotically AdS boundary conditions.…”

Section: Introductionmentioning

confidence: 96%

Connection versus metric description for non-AdS solutions in higher-spin theories

Lei

Ross

2015

Class. Quantum Grav.

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We consider recently-constructed solutions of three dimensional SL(N, R)×SL(N, R) Chern-Simons theories with non-relativistic symmetries. Solutions of the Chern-Simons theories can generically be mapped to solutions of a gravitational theory with a higherspin gauge symmetry. However, we will show that some of the non-relativistic solutions are not equivalent to metric solutions, as this mapping fails to be invertible. We also show that these Chern-Simons solutions always have a global SL(N, R) × SL(N, R) symmetry. We argue that these results pose a challenge to constructing a duality relating these solutions to field theories with non-relativistic symmetries.

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“…In refs. [18,19], some particular boundary conditions for higher spin gravity with gauge group SL(N, R) × SL(N, R) were associated to a generalized KdV hierarchy, and some particular cases, including the Boussinesq equation, were explicitly worked out. On the other hand, in ref.…”

Section: Introductionmentioning

confidence: 99%

Integrable systems and the boundary dynamics of higher spin gravity on AdS3

Ojeda

Pérez

2020

J. High Energ. Phys.

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We introduce a new set of boundary conditions for three-dimensional higher spin gravity with gauge group SL(3, ℝ) × SL(3, ℝ), where its dynamics at the boundary is described by the members of the modified Boussinesq integrable hierarchy. In the asymptotic region the gauge fields are written in the diagonal gauge, where the excitations go along the generators of the Cartan subalgebra of sl(3, ℝ) ⊕ sl(3, ℝ). We show that the entire integrable structure of the modified Boussinesq hierarchy, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are obtained from the asymptotic structure of the higher spin theory. Furthermore, its known relation with the Boussinesq hierarchy is inherited from our analysis once the asymptotic conditions are re-expressed in the highest weight gauge. Hence, the Miura map is recovered from a purely geometric construction in the bulk. Black holes that fit within our boundary conditions, the Hamiltonian reduction at the boundary, and the generalization to higher spin gravity with gauge group SL(N, ℝ) × SL(N, ℝ) are also discussed.

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Higher spin Lifshitz theories and the Korteweg-de Vries hierarchy

Cited by 17 publications

References 53 publications

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

Connection versus metric description for non-AdS solutions in higher-spin theories

Integrable systems and the boundary dynamics of higher spin gravity on AdS3

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