Integrable systems and the boundary dynamics of higher spin gravity on AdS 3
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Springer
Received: September 17, 2020 Accepted: October 10, 2020 Published: November 18, 2020
Emilio Ojedaa,b and Alfredo Péreza a
Centro de Estudios Científicos (CECs), Avenida Arturo Prat 514, Valdivia, Chile b Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
E-mail: [email protected], [email protected] Abstract: 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. Keywords: Higher Spin Gravity, Gauge-gravity correspondence, Black Holes ArXiv ePrint: 2009.07829
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)089
JHEP11(2020)089
Integrable systems and the boundary dynamics of higher spin gravity on AdS3
Contents 1 Introduction
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2 Review of the modified Boussinesq hierarchy
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4 Black holes
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5 Hamiltonian reduction and boundary dynamics
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6 Some extensions: Generalized Gibbs ensemble, spin-N gravity and modified Gelfand-Dickey hierarchies 16 6.1 Generalized Gibbs Ensemble 16 6.2 Higher spin gravity with gauge group SL (N, R) × SL (N, R) and modified Gelfand-Dickey hierarchies 17 7 Final remarks
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A Second Hamiltonian structure of the modified Boussinesq hierarchy
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B Gelfand-Dickey polynomials and Hamiltonians of the modified Boussinesq hierarchy 19 C Boussinesq hierarchy
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D Fundamental representation of the principal embedding of sl(2, R) within sl(N, R) 22 E Wess-Zumino term
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Introduction
The asymptotic structure of spacetime plays a fundamental role in the description of General Relativity in three dimensions. This theory does not possess local propagating degrees of freedom, and consequently, its dynamics is completely dictated by the choice of boundary conditions. In the case of a negative cosmological constant, it is standard practice the
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JHEP11(2020)089
3 Modified Boussinesq hierarchy from spin-3 gravity on AdS3 6 3.1 Chern-Simons formulation of spin-3 gravity on AdS3 6 3.2 Asymptotic behavior of the fields. Diagonal gauge 7 3.3 Boundary conditions fo
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