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Asymptotic Behaviour of Gravity in Three Dimensions Marc Henneaux

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Asymptotic Behaviour of Gravity in Three Dimensions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction Hamiltonian formulation and Asymptotic behaviour of gravity in three surface terms Asymptotic dimensions symmetries Algebra of charges Asymptotically anti-de Sitter Marc Henneaux spaces Central charge in 3 dimensions Chern-Simons reformulation IPMU, Tokyo, February 2011 Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Three-dimensional gravity, with or without cosmological constant, is at first sight trivial. Indeed, the Riemann tensor is completely determined by the Ricci tensor, and the Einstein equations imply therefore that the solutions are spaces of constant curvature, R ¤¡g g g g ¢ ¸¹½σ Æ ¸½ ¹σ ¡ λσ ¹½ (locally Minkowski, dS or AdS). There is no local degree of freedom. However, the case with negative cosmological constant has attracted in the last years a lot of interest. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction Hamiltonian formulation and surface terms Asymptotic symmetries Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Indeed, the Riemann tensor is completely determined by the Ricci tensor, and the Einstein equations imply therefore that the solutions are spaces of constant curvature, R ¤¡g g g g ¢ ¸¹½σ Æ ¸½ ¹σ ¡ λσ ¹½ (locally Minkowski, dS or AdS). There is no local degree of freedom. However, the case with negative cosmological constant has attracted in the last years a lot of interest. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Three-dimensional gravity, with or without cosmological Introduction constant, is at first sight trivial. Hamiltonian formulation and surface terms Asymptotic symmetries Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions However, the case with negative cosmological constant has attracted in the last years a lot of interest. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Three-dimensional gravity, with or without cosmological Introduction constant, is at first sight trivial. Hamiltonian formulation and Indeed, the Riemann tensor is completely determined by the surface terms Ricci tensor, and the Einstein equations imply therefore that the Asymptotic symmetries solutions are spaces of constant curvature, Algebra of charges Asymptotically ¡ ¢ anti-de Sitter R¸¹½σ ¤ g¸½g¹σ gλσg¹½ spaces Æ ¡ Central charge in (locally Minkowski, dS or AdS). There is no local degree of 3 dimensions Chern-Simons freedom. reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Three-dimensional gravity, with or without cosmological Introduction constant, is at first sight trivial. Hamiltonian formulation and Indeed, the Riemann tensor is completely determined by the surface terms Ricci tensor, and the Einstein equations imply therefore that the Asymptotic symmetries solutions are spaces of constant curvature, Algebra of charges Asymptotically ¡ ¢ anti-de Sitter R¸¹½σ ¤ g¸½g¹σ gλσg¹½ spaces Æ ¡ Central charge in (locally Minkowski, dS or AdS). There is no local degree of 3 dimensions Chern-Simons freedom. reformulation However, the case with negative cosmological constant has Supergravity in 3 dimensions attracted in the last years a lot of interest. Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions This is because even though the theory is locally trivial, it is not globally so. It admits black hole solutions which are obtained through identifications of anti-de-Sitter space. It has an interesting behaviour at infinity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . Æ 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction Hamiltonian formulation and surface terms Asymptotic symmetries Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions It admits black hole solutions which are obtained through identifications of anti-de-Sitter space. It has an interesting behaviour at infinity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . Æ 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction This is because even though the theory is locally trivial, it is not Hamiltonian globally so. formulation and surface terms Asymptotic symmetries Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions It has an interesting behaviour at infinity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . Æ 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction This is because even though the theory is locally trivial, it is not Hamiltonian globally so. formulation and surface terms It admits black hole solutions which are obtained through Asymptotic symmetries identifications of anti-de-Sitter space. Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction This is because even though the theory is locally trivial, it is not Hamiltonian globally so. formulation and surface terms It admits black hole solutions which are obtained through Asymptotic symmetries identifications of anti-de-Sitter space. Algebra of charges It has an interesting behaviour at infinity, where the asymptotic Asymptotically symmetry algebra is given by two copies of the Virasoro algebra anti-de Sitter spaces 3` with central charge c 2G . Central charge in Æ 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction This is because even though the theory is locally trivial, it is not Hamiltonian globally so. formulation and surface terms It admits black hole solutions which are obtained through Asymptotic symmetries identifications of anti-de-Sitter space. Algebra of charges It has an interesting behaviour at infinity, where the asymptotic Asymptotically symmetry algebra is given by two copies of the Virasoro algebra anti-de Sitter spaces 3` with central charge c 2G . Central charge in Æ 3 dimensions Three-dimensional gravity with negative cosmological constant Chern-Simons presents therefore many of the interesting features of the reformulation Supergravity in 3 higher-dimensional versions, but in a much simpler context. dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions The purpose of the talk is to explain the asymptotic structure of three-dimensional anti-de Sitter gravity. To that end, it is necessary to review the Hamiltonian approach to conserved charges in gravitation theory and understand the possible occurence of central charges. We shall first consider pure gravity. We shall then consider supersymmetric and higher spin extensions of 3D AdS gravity, using the Chern-Simons formulation. Introduction Asymptotic behaviour of gravity in three dimensions Marc Henneaux Introduction Hamiltonian formulation and surface terms Asymptotic symmetries Algebra of charges Asymptotically anti-de Sitter spaces Central charge in 3 dimensions Chern-Simons reformulation Supergravity in 3 dimensions Higher Spin Theory Conclusions Marc Henneaux Asymptotic behaviour of gravity in three dimensions To that end, it is necessary to review the Hamiltonian approach to conserved charges in gravitation theory and understand the possible occurence of central charges. We shall first consider pure gravity. We shall then consider supersymmetric and higher spin
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