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 September 2010 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 Λ¡g g g g ¢ λµρσ = λρ µσ − λσ µρ (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

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,

Introduction

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

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 Rλµρσ Λ gλρgµσ gλσgµρ spaces = − Central charge in (locally Minkowski, dS or AdS). There is no local degree of 3 dimensions

Chern-Simons freedom. reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction

Asymptotically ¡ ¢ anti-de Sitter Rλµρσ Λ gλρgµσ gλσgµρ spaces = − 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 Conclusions attracted in the last years a lot of interest.

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` . = 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

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` . = 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

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` . = 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

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 = 3 dimensions

Chern-Simons reformulation

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

Conclusions higher-dimensional versions, but in a much simpler context.

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 ﬁrst 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions To that end, it is necessary to ﬁrst 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 The purpose of the talk is to explain the asymptotic structure of Hamiltonian formulation and three-dimensional anti-de Sitter gravity. surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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 The purpose of the talk is to explain the asymptotic structure of Hamiltonian formulation and three-dimensional anti-de Sitter gravity. surface terms

Asymptotic To that end, it is necessary to ﬁrst review the Hamiltonian symmetries approach to conserved charges in gravitation theory and Algebra of charges understand the possible occurence of central charges. Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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 The purpose of the talk is to explain the asymptotic structure of Hamiltonian formulation and three-dimensional anti-de Sitter gravity. surface terms

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction The purpose of the talk is to explain the asymptotic structure of Hamiltonian formulation and three-dimensional anti-de Sitter gravity. surface terms

Central charge in We shall then consider supersymmetric and higher spin 3 dimensions extensions of 3D AdS gravity, using the Chern-Simons Chern-Simons reformulation formulation. Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Deﬁning conserved charges (energy, angular momentum, ...) in general relativity is a subtle issue. One approach is based on the Hamiltonian formulation of gravity. It has the advantage of clearly connecting the charges to the generators of the corresponding symmetries. It can be applied to asymptotic symmetries that are not exact symmetries. Finally, it gives direct access to the algebra of the charges and the possibility of non-Poissonian actions (“central charges")

Conserved charges and Hamiltonian formulation

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One approach is based on the Hamiltonian formulation of gravity. It has the advantage of clearly connecting the charges to the generators of the corresponding symmetries. It can be applied to asymptotic symmetries that are not exact symmetries. Finally, it gives direct access to the algebra of the charges and the possibility of non-Poissonian actions (“central charges")

Conserved charges and Hamiltonian formulation

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Deﬁning conserved charges (energy, angular momentum, ...) in

Hamiltonian general relativity is a subtle issue. formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It has the advantage of clearly connecting the charges to the generators of the corresponding symmetries. It can be applied to asymptotic symmetries that are not exact symmetries. Finally, it gives direct access to the algebra of the charges and the possibility of non-Poissonian actions (“central charges")

Conserved charges and Hamiltonian formulation

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Deﬁning conserved charges (energy, angular momentum, ...) in

Hamiltonian general relativity is a subtle issue. formulation and surface terms One approach is based on the Hamiltonian formulation of Asymptotic gravity. symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It can be applied to asymptotic symmetries that are not exact symmetries. Finally, it gives direct access to the algebra of the charges and the possibility of non-Poissonian actions (“central charges")

Conserved charges and Hamiltonian formulation

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Deﬁning conserved charges (energy, angular momentum, ...) in

Hamiltonian general relativity is a subtle issue. formulation and surface terms One approach is based on the Hamiltonian formulation of Asymptotic gravity. symmetries

Algebra of charges It has the advantage of clearly connecting the charges to the Asymptotically generators of the corresponding symmetries. anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Finally, it gives direct access to the algebra of the charges and the possibility of non-Poissonian actions (“central charges")

Conserved charges and Hamiltonian formulation

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Deﬁning conserved charges (energy, angular momentum, ...) in

Hamiltonian general relativity is a subtle issue. formulation and surface terms One approach is based on the Hamiltonian formulation of Asymptotic gravity. symmetries

Algebra of charges It has the advantage of clearly connecting the charges to the Asymptotically generators of the corresponding symmetries. anti-de Sitter spaces It can be applied to asymptotic symmetries that are not exact Central charge in 3 dimensions symmetries.

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Conserved charges and Hamiltonian formulation

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Deﬁning conserved charges (energy, angular momentum, ...) in

Hamiltonian general relativity is a subtle issue. formulation and surface terms One approach is based on the Hamiltonian formulation of Asymptotic gravity. symmetries

Chern-Simons reformulation Finally, it gives direct access to the algebra of the charges and the

Conclusions possibility of non-Poissonian actions (“central charges")

Marc Henneaux Asymptotic behaviour of gravity in three dimensions δ F [F,H[ξ]] ξ = where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) = R + with “bulk term" ddx(ξ H ξkH ) = Σ ⊥ + k and “surface term" H ( local expression) . = ∂Σ

Let F[g (x),πij(x), ] be a functional deﬁned on a spacelike ij ··· hypersurface Σ.

k ∂ Under an arbitrary surface deformation ξ ξ⊥n ξ of Σ, the = + ∂xk functional F changes as (Dirac, ADM)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

k ∂ Under an arbitrary surface deformation ξ ξ⊥n ξ of Σ, the = + ∂xk functional F changes as (Dirac, ADM)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Asymptotically anti-de Sitter k ∂ Under an arbitrary surface deformation ξ ξ⊥n ξ k of Σ, the spaces = + ∂x Central charge in functional F changes as (Dirac, ADM) 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) = R + with “bulk term" ddx(ξ H ξkH ) = Σ ⊥ + k and “surface term" H ( local expression) . = ∂Σ

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Asymptotically anti-de Sitter k ∂ Under an arbitrary surface deformation ξ ξ⊥n ξ k of Σ, the spaces = + ∂x Central charge in functional F changes as (Dirac, ADM) 3 dimensions δξF [F,H[ξ]] Chern-Simons = reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions R with “bulk term" ddx(ξ H ξkH ) = Σ ⊥ + k and “surface term" H ( local expression) . = ∂Σ

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Asymptotically anti-de Sitter k ∂ Under an arbitrary surface deformation ξ ξ⊥n ξ k of Σ, the spaces = + ∂x Central charge in functional F changes as (Dirac, ADM) 3 dimensions δ F [F,H[ξ]] ξ = Chern-Simons where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) reformulation = + Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions and “surface term" H ( local expression) . = ∂Σ

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux IDEA : conserved charges = generators of the corresponding

Introduction asymptotic symmetries in the Poisson bracket

Hamiltonian formulation and surface terms ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Asymptotic ··· symmetries hypersurface Σ.

Algebra of charges

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ij Dynamical variables : gij(x), π (x) µ ¶ 1 ij 1 2 1 1 H g− 2 π π π g 2 R 2Λg 2 = ij − d 1 − + −

j Hi 2π j = − i |

Standard general relativity :

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The bulk term vanishes on account of the constraints

Introduction H 0, Hk 0. Hamiltonian ≈ ≈ formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ij Dynamical variables : gij(x), π (x) µ ¶ 1 ij 1 2 1 1 H g− 2 π π π g 2 R 2Λg 2 = ij − d 1 − + −

j Hi 2π j = − i |

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The bulk term vanishes on account of the constraints

Introduction H 0, Hk 0. Hamiltonian ≈ ≈ formulation and surface terms

Asymptotic symmetries Standard general relativity : Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions µ ¶ 1 ij 1 2 1 1 H g− 2 π π π g 2 R 2Λg 2 = ij − d 1 − + −

j Hi 2π j = − i |

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The bulk term vanishes on account of the constraints

Introduction H 0, Hk 0. Hamiltonian ≈ ≈ formulation and surface terms

Asymptotic symmetries Standard general relativity : Algebra of charges ij Dynamical variables : gij(x), π (x) Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions j Hi 2π j = − i |

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The bulk term vanishes on account of the constraints

Introduction H 0, Hk 0. Hamiltonian ≈ ≈ formulation and surface terms

Asymptotic symmetries Standard general relativity : Algebra of charges ij Dynamical variables : gij(x), π (x) Asymptotically anti-de Sitter µ ¶ spaces 1 ij 1 2 1 1 H g− 2 πijπ π g 2 R 2Λg 2 Central charge in = − d 1 − + 3 dimensions − Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The bulk term vanishes on account of the constraints

Introduction H 0, Hk 0. Hamiltonian ≈ ≈ formulation and surface terms

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Requirement : H[ξ] should have well-deﬁned functional derivatives in the class of ﬁelds under consideration (“be differentiable") i.e., since δH[ξ] “Volume term" M[ξ] δQ[ξ] = + + where the volume term contains only undifferentiated variations

Z ³ ´ ddx Aij(x)δg (x) B (x)δπij(x) contributions from ij + ij + other ﬁelds

and where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form, one must impose M[ξ] δQ[ξ] 0. + =

Gravitational Hamiltonian

Asymptotic behaviour of How should one take the surface term Q[ξ] to be added to the gravity in three dimensions volume term ? 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions i.e., since δH[ξ] “Volume term" M[ξ] δQ[ξ] = + + where the volume term contains only undifferentiated variations

Z ³ ´ ddx Aij(x)δg (x) B (x)δπij(x) contributions from ij + ij + other ﬁelds

and where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form, one must impose M[ξ] δQ[ξ] 0. + =

Gravitational Hamiltonian

Asymptotic behaviour of How should one take the surface term Q[ξ] to be added to the gravity in three dimensions volume term ? Marc Henneaux Requirement : H[ξ] should have well-deﬁned functional Introduction derivatives in the class of ﬁelds under consideration (“be Hamiltonian formulation and differentiable") surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions where the volume term contains only undifferentiated variations

Z ³ ´ ddx Aij(x)δg (x) B (x)δπij(x) contributions from ij + ij + other ﬁelds

and where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form, one must impose M[ξ] δQ[ξ] 0. + =

Gravitational Hamiltonian

Asymptotic i.e., since δH[ξ] “Volume term" M[ξ] δQ[ξ] symmetries = + + Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions and where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form, one must impose M[ξ] δQ[ξ] 0. + =

Gravitational Hamiltonian

Asymptotic i.e., since δH[ξ] “Volume term" M[ξ] δQ[ξ] symmetries = + + Algebra of charges where the volume term contains only undifferentiated variations Asymptotically Z anti-de Sitter d ³ ij ij contributions from´ spaces d x A (x)δg (x) B (x)δπ (x) ij ij other ﬁelds Central charge in + + 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions one must impose M[ξ] δQ[ξ] 0. + =

Gravitational Hamiltonian

Chern-Simons and where M[ξ] is the surface term arising from integrations by reformulation parts to bring the volume term in desired form, Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Chern-Simons and where M[ξ] is the surface term arising from integrations by reformulation parts to bring the volume term in desired form, Conclusions one must impose M[ξ] δQ[ξ] 0. + =

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Z µ ¶ d ∂H ∂H δH[ξ] d x ξ⊥ δgij ξ⊥ δgij,m δQ[ξ] = ∂gij + ∂gij,m + ··· + Z "Ã µ ¶ ! # d ∂H ∂H d x ξ⊥ ξ⊥ δgij = ∂gij − ∂gij,m ,m + ··· + ··· Z µ ¶ d 1 ∂H d − Sm ξ⊥ δgij δQ[ξ] + ∂gij,m + ··· + | {z } M[ξ]

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux This can be seen as follows. Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Z "Ã µ ¶ ! # d ∂H ∂H d x ξ⊥ ξ⊥ δgij = ∂gij − ∂gij,m ,m + ··· + ··· Z µ ¶ d 1 ∂H d − Sm ξ⊥ δgij δQ[ξ] + ∂gij,m + ··· + | {z } M[ξ]

Gravitational Hamiltonian

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux This can be seen as follows. Introduction Hamiltonian Z µ ¶ formulation and d ∂H ∂H surface terms δH[ξ] d x ξ⊥ δgij ξ⊥ δgij,m δQ[ξ] = ∂gij + ∂gij,m + ··· + Asymptotic symmetries Z "Ã µ ¶ ! # Algebra of charges d ∂H ∂H d x ξ⊥ ξ⊥ δgij Asymptotically = ∂g − ∂g + ··· + ··· anti-de Sitter ij ij,m ,m spaces Z µ ¶ Central charge in d 1 ∂H 3 dimensions d − Sm ξ⊥ δgij δQ[ξ] + ∂gij,m + ··· + Chern-Simons reformulation | {z } M[ξ] Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One can then deﬁne the functional derivatives of H[ξ] as

δH[ξ] δH[ξ] Aij(x), B (x), etc. ij ij δgij(x) = δπ (x) =

The Hamiltonian action principle (pure gravity) µZ Z ¶ δ dt ddxπij g˙ dtH[ξ] 0 ij − =

yields under these conditions the correct equations of motion

δH[ξ] δH[ξ] δ g , δ πij . ξ ij ij ξ = δπ (x) = −δgij(x)

Gravitational Hamiltonian

Asymptotic behaviour of The condition M[ξ] δQ[ξ] 0 implies that the variation δH[ξ] gravity in three + = dimensions reduces to the volume term. 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The Hamiltonian action principle (pure gravity) µZ Z ¶ δ dt ddxπij g˙ dtH[ξ] 0 ij − =

yields under these conditions the correct equations of motion

δH[ξ] δH[ξ] δ g , δ πij . ξ ij ij ξ = δπ (x) = −δgij(x)

Gravitational Hamiltonian

Asymptotic behaviour of The condition M[ξ] δQ[ξ] 0 implies that the variation δH[ξ] gravity in three + = dimensions reduces to the volume term. Marc Henneaux One can then deﬁne the functional derivatives of H[ξ] as Introduction Hamiltonian δH[ξ] δH[ξ] formulation and Aij(x), B (x), etc. surface terms ij ij δgij(x) = δπ (x) = Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions δH[ξ] δH[ξ] δ g , δ πij . ξ ij ij ξ = δπ (x) = −δgij(x)

Gravitational Hamiltonian

Algebra of charges The Hamiltonian action principle (pure gravity) Asymptotically anti-de Sitter spaces µZ Z ¶ d ij Central charge in δ dt d xπ g˙ij dtH[ξ] 0 3 dimensions − = Chern-Simons reformulation yields under these conditions the correct equations of motion Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The surface term M[ξ] picked up upon integration by parts can be worked out. For instance, for pure gravity, it takes the universal form (independently of the boundary conditions and of the spacetime dimension), Z d ijmn ¡ ¢ M[ξ] d SnG ξ⊥δgij m ξ,⊥mδgij = − | − Z ddS 2N δπmn − n m Z ddS (2Nkπmn Nnπkm)δg . − n − km

If one modiﬁes the Lagrangian (by the topological mass term in 3 dimensions, or by matter ﬁelds, or in any other (local) way), the explicit expression for M[ξ] changes but can again be worked out.

Boundary conditions

Asymptotic behaviour of Analysis of the condition M[ξ] δQ[ξ] 0 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions For instance, for pure gravity, it takes the universal form (independently of the boundary conditions and of the spacetime dimension), Z d ijmn ¡ ¢ M[ξ] d SnG ξ⊥δgij m ξ,⊥mδgij = − | − Z ddS 2N δπmn − n m Z ddS (2Nkπmn Nnπkm)δg . − n − km

Boundary conditions

Asymptotic behaviour of Analysis of the condition M[ξ] δQ[ξ] 0 gravity in three + = dimensions The surface term M[ξ] picked up upon integration by parts can Marc Henneaux be worked out. Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions If one modiﬁes the Lagrangian (by the topological mass term in 3 dimensions, or by matter ﬁelds, or in any other (local) way), the explicit expression for M[ξ] changes but can again be worked out.

Boundary conditions

Hamiltonian For instance, for pure gravity, it takes the universal form formulation and surface terms (independently of the boundary conditions and of the spacetime

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions

Hamiltonian For instance, for pure gravity, it takes the universal form formulation and surface terms (independently of the boundary conditions and of the spacetime

Asymptotic dimension), symmetries Z Algebra of charges d ijmn ¡ ¢ M[ξ] d SnG ξ⊥δgij m ξ,⊥mδgij Asymptotically | anti-de Sitter = − − spaces Z d mn Central charge in d Sn2Nmδπ 3 dimensions − Z Chern-Simons d k mn n km reformulation d S (2N π N π )δg . − n − km Conclusions If one modiﬁes the Lagrangian (by the topological mass term in 3 dimensions, or by matter ﬁelds, or in any other (local) way), the explicit expression for M[ξ] changes but can again be worked out.

Marc Henneaux Asymptotic behaviour of gravity in three dimensions In general, the requirement M[ξ] δQ[ξ] 0 has no solution + = because M[ξ] is not integrable (i.e., not the variation of a local surface term). Hence one needs to restrict the allowed class of ﬁelds by boundary conditions. Thus, even though the expression for M[ξ] can be worked out without knowing the boundary conditions, verifying explicitly the existence of Q[ξ] requires a speciﬁc form of the boundary conditions.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ]

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Hence one needs to restrict the allowed class of ﬁelds by boundary conditions. Thus, even though the expression for M[ξ] can be worked out without knowing the boundary conditions, verifying explicitly the existence of Q[ξ] requires a speciﬁc form of the boundary conditions.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian In general, the requirement M[ξ] δQ[ξ] 0 has no solution formulation and + = surface terms because M[ξ] is not integrable (i.e., not the variation of a local Asymptotic surface term). symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Thus, even though the expression for M[ξ] can be worked out without knowing the boundary conditions, verifying explicitly the existence of Q[ξ] requires a speciﬁc form of the boundary conditions.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Algebra of charges Hence one needs to restrict the allowed class of ﬁelds by Asymptotically boundary conditions. anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Algebra of charges Hence one needs to restrict the allowed class of ﬁelds by Asymptotically boundary conditions. anti-de Sitter spaces Thus, even though the expression for M[ξ] can be worked out Central charge in 3 dimensions without knowing the boundary conditions, verifying explicitly Chern-Simons the existence of Q[ξ] requires a speciﬁc form of the boundary reformulation conditions. Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The boundary conditions should make M[ξ] integrable in field space, i.e., equal to the variation of a surface term. When this condition is met, Q[ξ] is defined by the equation M[ξ] δQ[ξ] 0 up to a constant C[ξ]. + = This constant is usually taken to be zero for some natural “background" configuration. Which boundary conditions to impose depends on the context.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions When this condition is met, Q[ξ] is deﬁned by the equation M[ξ] δQ[ξ] 0 up to a constant C[ξ]. + = This constant is usually taken to be zero for some natural “background" conﬁguration. Which boundary conditions to impose depends on the context.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian formulation and The boundary conditions should make M[ξ] integrable in ﬁeld surface terms

Asymptotic space, i.e., equal to the variation of a surface term. symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This constant is usually taken to be zero for some natural “background" conﬁguration. Which boundary conditions to impose depends on the context.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian formulation and The boundary conditions should make M[ξ] integrable in ﬁeld surface terms

Asymptotic space, i.e., equal to the variation of a surface term. symmetries When this condition is met, Q[ξ] is deﬁned by the equation Algebra of charges

Asymptotically M[ξ] δQ[ξ] 0 up to a constant C[ξ]. anti-de Sitter + = spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Which boundary conditions to impose depends on the context.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian formulation and The boundary conditions should make M[ξ] integrable in ﬁeld surface terms

Asymptotic space, i.e., equal to the variation of a surface term. symmetries When this condition is met, Q[ξ] is deﬁned by the equation Algebra of charges

Asymptotically M[ξ] δQ[ξ] 0 up to a constant C[ξ]. anti-de Sitter + = spaces This constant is usually taken to be zero for some natural Central charge in “background" conﬁguration. 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Determination of the surface term Q[ξ] Hamiltonian formulation and The boundary conditions should make M[ξ] integrable in ﬁeld surface terms

Asymptotic space, i.e., equal to the variation of a surface term. symmetries When this condition is met, Q[ξ] is deﬁned by the equation Algebra of charges

Chern-Simons Which boundary conditions to impose depends on the context. reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetries are by deﬁnition the surface k deformations ξ (ξ⊥,ξ ) that preserve the boundary conditions, = modulo the “trivial" ones to be deﬁned below. The generator of an asymptotic symmetry reads Z d ³ k ´ H[ξ] d x ξ⊥H ξ H Q[ξ] = + k +

and reduces on shell to the surface term. Q[ξ] H[ξ] is the conserved charge associated with the ≈ asymptotic symmetry. It depends only on the asymptotic form of ξ. Two asymptotic symmetries that have the same asymptotic behaviour have the same charge and should be identiﬁed.

Asymptotic symmetries - Deﬁnition and properties

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The generator of an asymptotic symmetry reads Z d ³ k ´ H[ξ] d x ξ⊥H ξ H Q[ξ] = + k +

Asymptotic symmetries - Deﬁnition and properties

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Q[ξ] H[ξ] is the conserved charge associated with the ≈ asymptotic symmetry. It depends only on the asymptotic form of ξ. Two asymptotic symmetries that have the same asymptotic behaviour have the same charge and should be identiﬁed.

Asymptotic symmetries - Deﬁnition and properties

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Two asymptotic symmetries that have the same asymptotic behaviour have the same charge and should be identiﬁed.

Asymptotic symmetries - Deﬁnition and properties

Asymptotic behaviour of gravity in three dimensions The asymptotic symmetries are by deﬁnition the surface Marc Henneaux k deformations ξ (ξ⊥,ξ ) that preserve the boundary conditions, Introduction = modulo the “trivial" ones to be deﬁned below. Hamiltonian formulation and surface terms The generator of an asymptotic symmetry reads Asymptotic Z symmetries d ³ k ´ H[ξ] d x ξ⊥H ξ Hk Q[ξ] Algebra of charges = + + Asymptotically anti-de Sitter spaces and reduces on shell to the surface term. Central charge in Q[ξ] H[ξ] is the conserved charge associated with the 3 dimensions ≈ Chern-Simons asymptotic symmetry. It depends only on the asymptotic form of reformulation ξ. Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotic symmetries - Deﬁnition and properties

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetry algebra is the quotient algebra of the asymptotic symmetries by the “trivial" asymptotic symmetries with zero charge. The quotient is well deﬁned because the trivial asymptotic symmetries form an ideal,

δ Q[ξ] 0 η = when η 0 at (Q[ξ] is “gauge-invariant"). → ∞ Note that the asymptotic symmetries may not always be realized as background isometries. The asymptotic symmetries with zero charges are the true gauge symmetries.

Asymptotic symmetry algebra

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The quotient is well deﬁned because the trivial asymptotic symmetries form an ideal,

Asymptotic symmetry algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry algebra is the quotient algebra of the

Introduction asymptotic symmetries by the “trivial" asymptotic symmetries

Hamiltonian with zero charge. formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Note that the asymptotic symmetries may not always be realized as background isometries. The asymptotic symmetries with zero charges are the true gauge symmetries.

Asymptotic symmetry algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry algebra is the quotient algebra of the

Introduction asymptotic symmetries by the “trivial" asymptotic symmetries

Hamiltonian with zero charge. formulation and surface terms The quotient is well deﬁned because the trivial asymptotic Asymptotic symmetries form an ideal, symmetries

Algebra of charges

Asymptotically δηQ[ξ] 0 anti-de Sitter = spaces when η 0 at (Q[ξ] is “gauge-invariant"). Central charge in → ∞ 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetries with zero charges are the true gauge symmetries.

Asymptotic symmetry algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry algebra is the quotient algebra of the

Introduction asymptotic symmetries by the “trivial" asymptotic symmetries

Hamiltonian with zero charge. formulation and surface terms The quotient is well deﬁned because the trivial asymptotic Asymptotic symmetries form an ideal, symmetries

Algebra of charges

Asymptotically δηQ[ξ] 0 anti-de Sitter = spaces when η 0 at (Q[ξ] is “gauge-invariant"). Central charge in → ∞ 3 dimensions Note that the asymptotic symmetries may not always be realized Chern-Simons reformulation as background isometries.

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotic symmetry algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry algebra is the quotient algebra of the

Introduction asymptotic symmetries by the “trivial" asymptotic symmetries

Hamiltonian with zero charge. formulation and surface terms The quotient is well deﬁned because the trivial asymptotic Asymptotic symmetries form an ideal, symmetries

Algebra of charges

Theorem :

Hence [H[ξ],H[η]] and H £[ξ,η]¤ differ, up to trivial terms, by a c-number K [ξ,η] (which fulﬁlls cocycle condition etc...).

Algebra of charges : A useful theorem

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction If ξ and η are asymptotic symmetries, then their commutator [ξ,η]

Hamiltonian also is. Furthermore, one has the following useful theorem formulation and surface terms (Brown-Henneaux JMP 1986)

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions [H[ξ],H[η]] and H £[ξ,η]¤ generate the same asymptotic transformation i.e.,[F,[H[ξ],H[η]]] [F,H £[ξ,η]¤] for any functional F of the = dynamical variables. Hence [H[ξ],H[η]] and H £[ξ,η]¤ differ, up to trivial terms, by a c-number K [ξ,η] (which fulﬁlls cocycle condition etc...).

Algebra of charges : A useful theorem

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction If ξ and η are asymptotic symmetries, then their commutator [ξ,η]

Hamiltonian also is. Furthermore, one has the following useful theorem formulation and surface terms (Brown-Henneaux JMP 1986) Asymptotic Theorem : symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions i.e.,[F,[H[ξ],H[η]]] [F,H £[ξ,η]¤] for any functional F of the = dynamical variables. Hence [H[ξ],H[η]] and H £[ξ,η]¤ differ, up to trivial terms, by a c-number K [ξ,η] (which fulﬁlls cocycle condition etc...).

Algebra of charges : A useful theorem

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction If ξ and η are asymptotic symmetries, then their commutator [ξ,η]

Hamiltonian also is. Furthermore, one has the following useful theorem formulation and surface terms (Brown-Henneaux JMP 1986) Asymptotic Theorem : symmetries £ ¤ Algebra of charges [H[ξ],H[η]] and H [ξ,η] generate the same asymptotic

Asymptotically transformation anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Hence [H[ξ],H[η]] and H £[ξ,η]¤ differ, up to trivial terms, by a c-number K [ξ,η] (which fulﬁlls cocycle condition etc...).

Algebra of charges : A useful theorem

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction If ξ and η are asymptotic symmetries, then their commutator [ξ,η]

Asymptotically transformation anti-de Sitter i.e.,[F,[H[ξ],H[η]]] [F,H £[ξ,η]¤] for any functional F of the spaces = Central charge in dynamical variables. 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Algebra of charges : A useful theorem

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction If ξ and η are asymptotic symmetries, then their commutator [ξ,η]

Asymptotically transformation anti-de Sitter i.e.,[F,[H[ξ],H[η]]] [F,H £[ξ,η]¤] for any functional F of the spaces = Central charge in dynamical variables. 3 dimensions Hence [H[ξ],H[η]] and H £[ξ,η]¤ differ, up to trivial terms, by a Chern-Simons reformulation c-number K [ξ,η] (which fulﬁlls cocycle condition etc...). Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The cohomology group H2(G ) must be different from zero. The asymptotic symmetry group cannot be realized as background isometry group.

Deﬁne H H[ξ ] where ξ is a basis of the asymptotic A = A A symmetry algebra (HA deﬁned up to trivial terms). Then [H ,H ] f C H K (up to trivial terms) A B = AB C + AB which implies [Q ,Q ] f C Q K (in the Dirac bracket). A B = AB C + AB When can KAB be different from zero ?

Expression in a basis

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

Conclusions

Then [H ,H ] f C H K (up to trivial terms) A B = AB C + AB which implies [Q ,Q ] f C Q K (in the Dirac bracket). A B = AB C + AB When can KAB be different from zero ?

Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian Deﬁne HA H[ξA] where ξA is a basis of the asymptotic formulation and = surface terms symmetry algebra (HA deﬁned up to trivial terms).

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

which implies [Q ,Q ] f C Q K (in the Dirac bracket). A B = AB C + AB When can KAB be different from zero ?

Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

When can KAB be different from zero ?

Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian Deﬁne HA H[ξA] where ξA is a basis of the asymptotic formulation and = surface terms symmetry algebra (HA deﬁned up to trivial terms). C Asymptotic Then [HA,HB] f HC KAB (up to trivial terms) symmetries = AB + C Algebra of charges which implies [Q ,Q ] f Q K (in the Dirac bracket). A B = AB C + AB Asymptotically anti-de Sitter When can KAB be different from zero ? spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetry group cannot be realized as background isometry group.

Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Expression in a basis

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Chern-Simons isometry group. reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Theorem : If the asymptotic symmetry group can be realized as isometry group of some background, then K 0. AB = Proof : Adjust the constant in Q so that Q 0 for the background. A A = Next, observe that the relation [Q ,Q ] f C Q K can be read A B = AB C + AB δ Q f C Q K . B A = AB C + AB

Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions If the asymptotic symmetry group can be realized as isometry group of some background, then K 0. AB = Proof : Adjust the constant in Q so that Q 0 for the background. A A = Next, observe that the relation [Q ,Q ] f C Q K can be read A B = AB C + AB δ Q f C Q K . B A = AB C + AB

Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Asymptotic behaviour of gravity in three Theorem : 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Proof : Adjust the constant in Q so that Q 0 for the background. A A = Next, observe that the relation [Q ,Q ] f C Q K can be read A B = AB C + AB δ Q f C Q K . B A = AB C + AB

Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Adjust the constant in Q so that Q 0 for the background. A A = Next, observe that the relation [Q ,Q ] f C Q K can be read A B = AB C + AB δ Q f C Q K . B A = AB C + AB

Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Next, observe that the relation [Q ,Q ] f C Q K can be read A B = AB C + AB δ Q f C Q K . B A = AB C + AB

Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Asymptotic Adjust the constant in QA so that QA 0 for the background. symmetries =

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Evaluated on the background, this relation reduces to

δ Q K . B A = AB

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Asymptotic behaviour of gravity in three Theorem : dimensions Marc Henneaux If the asymptotic symmetry group can be realized as isometry group of some background, then KAB 0. Introduction = Hamiltonian formulation and Proof : surface terms Adjust the constant in Q so that Q 0 for the background. Asymptotic A A = symmetries C Next, observe that the relation [QA,QB] f QC KAB can be read Algebra of charges = AB + Asymptotically anti-de Sitter C δBQA f AB QC KAB. spaces = + Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB =

Vanishing central charges

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Vanishing central charges

If the background is invariant (strictly and not just asymptotically), δ Q 0 and hence, K 0. B A = AB = Marc Henneaux Asymptotic behaviour of gravity in three dimensions N 2 supergravity in 4 dimensions with asymptotically ﬂat = boundary conditions. The asymptotic symmetry algebra is the N 2 super-Poincaré = algebra times U(1) (electric). In the presence of a magnetic pole, this algebra acquires a non-vanishing central charge (U(1) magnetic). AdS gravity in 3 dimensions. This theory is discussed below.

Non-vanishing central charges

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Theories exist, however, with K 0 : Introduction AB 6= Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions AdS gravity in 3 dimensions. This theory is discussed below.

Non-vanishing central charges

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Theories exist, however, with K 0 : Introduction AB 6= Hamiltonian formulation and surface terms N 2 supergravity in 4 dimensions with asymptotically ﬂat = Asymptotic boundary conditions. symmetries The asymptotic symmetry algebra is the N 2 super-Poincaré Algebra of charges = algebra times U(1) (electric). In the presence of a magnetic pole, Asymptotically anti-de Sitter this algebra acquires a non-vanishing central charge (U(1) spaces

Central charge in magnetic). 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Non-vanishing central charges

Central charge in magnetic). 3 dimensions AdS gravity in 3 dimensions. Chern-Simons reformulation This theory is discussed below. Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The background metric is (the universal cover of) AdS,

³ ³ r ´´ ³ ³ r ´´ 1 ds2 1 dt2 1 − dr2 r2 dω2, = − + l + + l +

D 3 : dω2 dφ2, D 4 : dω2 dθ2 sin2 θdφ2, etc = = = = + with D d 1. = + Maximally symmetric space with isometry group O(D 1,2). − D 3 : O(2,2) ; D 4 : O(3,2) etc. = =

Asymptotically anti-de Sitter spaces - Background metric

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Want to describe isolated systems in an AdS background Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Maximally symmetric space with isometry group O(D 1,2). − D 3 : O(2,2) ; D 4 : O(3,2) etc. = =

Asymptotically anti-de Sitter spaces - Background metric

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Want to describe isolated systems in an AdS background Introduction

Hamiltonian The background metric is (the universal cover of) AdS, formulation and surface terms 1 2 ³ ³ r ´´ 2 ³ ³ r ´´− 2 2 2 Asymptotic ds 1 dt 1 dr r dω , symmetries = − + l + + l + Algebra of charges

Asymptotically 2 2 2 2 2 2 anti-de Sitter D 3 : dω dφ , D 4 : dω dθ sin θdφ , etc spaces = = = = + Central charge in with D d 1. 3 dimensions = + Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions D 3 : O(2,2) ; D 4 : O(3,2) etc. = =

Asymptotically anti-de Sitter spaces - Background metric

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Want to describe isolated systems in an AdS background Introduction

Asymptotically 2 2 2 2 2 2 anti-de Sitter D 3 : dω dφ , D 4 : dω dθ sin θdφ , etc spaces = = = = + Central charge in with D d 1. 3 dimensions = + Chern-Simons Maximally symmetric space with isometry group O(D 1,2). reformulation − Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotically anti-de Sitter spaces - Background metric

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Want to describe isolated systems in an AdS background Introduction

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Following Penrose, one expects the asymptotic symmetry group to be the conformal group of the boundary. Asymptotically, the metric behaves as

2 2 2 2 2 2 2 ds r dt r− dr r dω → − + + · µ dr ¶2¸ r2 dt2 dω2 . = − + + r2

The redefinition of the radial coordinate 1 r , dr r2dρ = 1 ρ = − brings the boundary at infinity at the finite value ρ 1. =

Boundary of anti-de Sitter space

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotically, the metric behaves as

2 2 2 2 2 2 2 ds r dt r− dr r dω → − + + · µ dr ¶2¸ r2 dt2 dω2 . = − + + r2

The redefinition of the radial coordinate 1 r , dr r2dρ = 1 ρ = − brings the boundary at infinity at the finite value ρ 1. =

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions Following Penrose, one expects the asymptotic symmetry group Marc Henneaux to be the conformal group of the boundary. Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The redefinition of the radial coordinate 1 r , dr r2dρ = 1 ρ = − brings the boundary at infinity at the finite value ρ 1. =

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions Following Penrose, one expects the asymptotic symmetry group Marc Henneaux to be the conformal group of the boundary. Introduction

Hamiltonian Asymptotically, the metric behaves as formulation and surface terms 2 2 2 2 2 2 2 Asymptotic ds r dt r− dr r dω symmetries → − · + +µ ¶2¸ 2 2 2 dr Algebra of charges r dt dω . = − + + r2 Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions Following Penrose, one expects the asymptotic symmetry group Marc Henneaux to be the conformal group of the boundary. Introduction

Central charge in 3 dimensions The redefinition of the radial coordinate Chern-Simons 1 reformulation r , dr r2dρ Conclusions = 1 ρ = − brings the boundary at infinity at the finite value ρ 1. =

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The AdS metric reads ds2 Ω2dσ2 = with Ω r and = dσ2 dt2 dω2 dρ2 . = − + +

d 1 The boundary at ρ 1 is a timelike surface with topology R S − . = × AdS isometries are conformal transformations of the boundary - and all of them for D 4. ≥ For D 3, the boundary is the cylinder R S1 with metric = × dt2 dφ2 and the conformal group at infinity is − + infinite-dimensional and contains O(2,2), which is a proper finite-dimensional subgroup.

Boundary of anti-de Sitter space

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions d 1 The boundary at ρ 1 is a timelike surface with topology R S − . = × AdS isometries are conformal transformations of the boundary - and all of them for D 4. ≥ For D 3, the boundary is the cylinder R S1 with metric = × dt2 dφ2 and the conformal group at infinity is − + infinite-dimensional and contains O(2,2), which is a proper finite-dimensional subgroup.

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The AdS metric reads ds2 Ω2dσ2 Introduction = Hamiltonian with Ω r and formulation and = surface terms dσ2 dt2 dω2 dρ2 . Asymptotic = − + + symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions AdS isometries are conformal transformations of the boundary - and all of them for D 4. ≥ For D 3, the boundary is the cylinder R S1 with metric = × dt2 dφ2 and the conformal group at infinity is − + infinite-dimensional and contains O(2,2), which is a proper finite-dimensional subgroup.

Boundary of anti-de Sitter space

Algebra of charges d 1 Asymptotically The boundary at ρ 1 is a timelike surface with topology R S − . anti-de Sitter = × spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions For D 3, the boundary is the cylinder R S1 with metric = × dt2 dφ2 and the conformal group at infinity is − + infinite-dimensional and contains O(2,2), which is a proper finite-dimensional subgroup.

Boundary of anti-de Sitter space

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary of anti-de Sitter space

Algebra of charges d 1 Asymptotically The boundary at ρ 1 is a timelike surface with topology R S − . anti-de Sitter = × spaces AdS isometries are conformal transformations of the boundary - Central charge in and all of them for D 4. 3 dimensions ≥ 1 Chern-Simons For D 3, the boundary is the cylinder R S with metric reformulation = × dt2 dφ2 and the conformal group at infinity is Conclusions − + infinite-dimensional and contains O(2,2), which is a proper finite-dimensional subgroup.

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The situation is thus the following :

Background Conformal group isometries at inﬁnity D 4 O(D 1,2) O(D 1,2) ≥ − − D 3 O(2,2) C (2) O(2,2) = ⊃

One thus expects that the AdS group O(D 1,2) is the asymptotic − symmetry group in D 4 dimensions and that the ≥ inﬁnite-dimensional conformal group C (2) is the asymptotic symmetry group in D 3. In that latter case, central charges are = allowed and do, in fact, occur. The question is to explicitly give the precise boundary conditions that implement these general ideas.

Boundary of anti-de Sitter space

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Background Conformal group isometries at inﬁnity D 4 O(D 1,2) O(D 1,2) ≥ − − D 3 O(2,2) C (2) O(2,2) = ⊃

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions The situation is thus the following : 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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One thus expects that the AdS group O(D 1,2) is the asymptotic − symmetry group in D 4 dimensions and that the ≥ inﬁnite-dimensional conformal group C (2) is the asymptotic symmetry group in D 3. In that latter case, central charges are = allowed and do, in fact, occur. The question is to explicitly give the precise boundary conditions that implement these general ideas.

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions The situation is thus the following : Marc Henneaux

Introduction Background Conformal group Hamiltonian formulation and isometries at inﬁnity surface terms D 4 O(D 1,2) O(D 1,2) Asymptotic ≥ − − symmetries D 3 O(2,2) C (2) O(2,2) Algebra of charges = ⊃

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The question is to explicitly give the precise boundary conditions that implement these general ideas.

Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions The situation is thus the following : Marc Henneaux

Asymptotically anti-de Sitter One thus expects that the AdS group O(D 1,2) is the asymptotic spaces − symmetry group in D 4 dimensions and that the Central charge in ≥ 3 dimensions inﬁnite-dimensional conformal group C (2) is the asymptotic Chern-Simons symmetry group in D 3. In that latter case, central charges are reformulation = Conclusions allowed and do, in fact, occur.

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary of anti-de Sitter space

Asymptotic behaviour of gravity in three dimensions The situation is thus the following : Marc Henneaux

Marc Henneaux Asymptotic behaviour of gravity in three dimensions They should contain the physically relevant solutions. They should make the surface integrals ﬁnite and integrable. They should be such that the asymptotic symmetry group contains the AdS group.

The boundary conditions depend on the theory.

They should fulﬁll the following requirements :

Boundary conditions fulfilling these properties have been devised for D 4 in M.H.+ C. Teitelboim 1985 (see also MH 1985) ≥ for pure gravity or gravity coupled to “fastly decaying" matter fields. They have been extended to include “slowly decaying" scalar fields in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2004, 2007).

Boundary conditions

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

Conclusions

They should fulﬁll the following requirements :

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. 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

Conclusions

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Introduction They should fulﬁll the following requirements : Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions They should make the surface integrals ﬁnite and integrable. They should be such that the asymptotic symmetry group contains the AdS group.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Introduction They should fulﬁll the following requirements : Hamiltonian formulation and They should contain the physically relevant solutions. surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions They should be such that the asymptotic symmetry group contains the AdS group.

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Introduction They should fulﬁll the following requirements : Hamiltonian formulation and They should contain the physically relevant solutions. surface terms They should make the surface integrals ﬁnite and integrable. Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions fulfilling these properties have been devised for D 4 in M.H.+ C. Teitelboim 1985 (see also MH 1985) ≥ for pure gravity or gravity coupled to “fastly decaying" matter fields. They have been extended to include “slowly decaying" scalar fields in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2004, 2007).

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions They have been extended to include “slowly decaying" scalar ﬁelds in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2004, 2007).

Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions

Asymptotic behaviour of gravity in three dimensions The boundary conditions depend on the theory. Marc Henneaux

Asymptotically anti-de Sitter spaces Boundary conditions fulfilling these properties have been Central charge in devised for D 4 in M.H.+ C. Teitelboim 1985 (see also MH 1985) 3 dimensions ≥ Chern-Simons for pure gravity or gravity coupled to “fastly decaying" matter reformulation fields. Conclusions They have been extended to include “slowly decaying" scalar fields in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2004, 2007).

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetry group is the anti-de Sitter group O(D 1,2). − The Hamiltonian generator reads Z d ³ k ´ 1 AB H[ξ] d x ξ⊥H ξ Hk ξ QAB = + + 2 ∞

where the surface deformation ξ behaves as ξ 1 ξABη for → 2 AB r , with η the AdS Killing vectors. ∞ → ∞ AB The surface integrals, adjusted so that Q (AdS) 0 fulﬁll the AB = so(D 1,2) algebra without central charge according to the − general theorems. One recovers in particular the Abbott-Deser energy (1982).

Surface Integrals (D 4) ≥

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The Hamiltonian generator reads Z d ³ k ´ 1 AB H[ξ] d x ξ⊥H ξ Hk ξ QAB = + + 2 ∞

Surface Integrals (D 4) ≥

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry group is the anti-de Sitter group O(D 1,2). Introduction − Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The surface integrals, adjusted so that Q (AdS) 0 fulﬁll the AB = so(D 1,2) algebra without central charge according to the − general theorems. One recovers in particular the Abbott-Deser energy (1982).

Surface Integrals (D 4) ≥

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One recovers in particular the Abbott-Deser energy (1982).

Surface Integrals (D 4) ≥

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The asymptotic symmetry group is the anti-de Sitter group O(D 1,2). Introduction − Hamiltonian The Hamiltonian generator reads formulation and surface terms Z Asymptotic d ³ k ´ 1 AB symmetries H[ξ] d x ξ⊥H ξ Hk ξ QAB = + + 2 ∞ Algebra of charges Asymptotically 1 AB anti-de Sitter where the surface deformation ξ behaves as ξ 2 ξ ηAB for spaces → ∞ r , with ηAB the AdS Killing vectors. Central charge in → ∞ 3 dimensions The surface integrals, adjusted so that QAB(AdS) 0 fulﬁll the Chern-Simons = so(D 1,2) algebra without central charge according to the reformulation − Conclusions general theorems.

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Surface Integrals (D 4) ≥

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The D 3 case was investigated in M.H.+ David Brown (1986) for = pure gravity or gravity coupled to “fastly decaying" matter ﬁelds. The analysis has been extended to include “slowly decaying" scalar ﬁelds in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2002), and to include a topological mass term in MH-C.Martínez-R.Troncoso 2009) In the coordinate system where the background metric reads µ 2 ¶ 1 2 µ 2 ¶ r − 2 l ¡ 2 2¢ l 2 1 dr (dx+) (dx−) r dx+ dx−, + l2 − 4 + − 2 +

one allows (in the ﬁrst case) deviations ∆gαβ from AdS that behave as

frr 5 frm 4 ∆grr O(r− ), ∆grm O(r− ), = r4 + = r3 + 1 ∆g f O(r− ), mn = mn + with m , and f (xm). = + − αβ

D 3 - Boundary conditions =

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The analysis has been extended to include “slowly decaying" scalar ﬁelds in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2002), and to include a topological mass term in MH-C.Martínez-R.Troncoso 2009) In the coordinate system where the background metric reads µ 2 ¶ 1 2 µ 2 ¶ r − 2 l ¡ 2 2¢ l 2 1 dr (dx+) (dx−) r dx+ dx−, + l2 − 4 + − 2 +

one allows (in the ﬁrst case) deviations ∆gαβ from AdS that behave as

frr 5 frm 4 ∆grr O(r− ), ∆grm O(r− ), = r4 + = r3 + 1 ∆g f O(r− ), mn = mn + with m , and f (xm). = + − αβ

D 3 - Boundary conditions =

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions In the coordinate system where the background metric reads µ 2 ¶ 1 2 µ 2 ¶ r − 2 l ¡ 2 2¢ l 2 1 dr (dx+) (dx−) r dx+ dx−, + l2 − 4 + − 2 +

one allows (in the ﬁrst case) deviations ∆gαβ from AdS that behave as

frr 5 frm 4 ∆grr O(r− ), ∆grm O(r− ), = r4 + = r3 + 1 ∆g f O(r− ), mn = mn + with m , and f (xm). = + − αβ

D 3 - Boundary conditions =

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions one allows (in the ﬁrst case) deviations ∆gαβ from AdS that behave as

frr 5 frm 4 ∆grr O(r− ), ∆grm O(r− ), = r4 + = r3 + 1 ∆g f O(r− ), mn = mn + with m , and f (xm). = + − αβ

D 3 - Boundary conditions =

Asymptotic behaviour of The D 3 case was investigated in M.H.+ David Brown (1986) for gravity in three = dimensions pure gravity or gravity coupled to “fastly decaying" matter ﬁelds. Marc Henneaux The analysis has been extended to include “slowly decaying" Introduction scalar ﬁelds in M.H. + C. Martínez + R. Troncoso + J. Zanelli Hamiltonian formulation and (2002), and to include a topological mass term in surface terms MH-C.Martínez-R.Troncoso 2009) Asymptotic symmetries In the coordinate system where the background metric reads Algebra of charges µ 2 ¶ 1 2 µ 2 ¶ r − 2 l ¡ 2 2¢ l 2 Asymptotically 1 dr (dx+) (dx−) r dx+ dx−, anti-de Sitter + l2 − 4 + − 2 + spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions D 3 - Boundary conditions =

Central charge in 3 dimensions one allows (in the ﬁrst case) deviations ∆gαβ from AdS that Chern-Simons reformulation behave as Conclusions frr 5 frm 4 ∆grr O(r− ), ∆grm O(r− ), = r4 + = r3 + 1 ∆g f O(r− ), mn = mn + with m , and f (xm). = + − αβ Marc Henneaux Asymptotic behaviour of gravity in three dimensions These boundary conditions are invariant under diffeomorphisms ηα of the form

2 l 2 η+ T + 2 ∂ T − , = + 2r − + ··· 2 l 2 η− T − 2 ∂ T + , = + 2r + + ··· r r ¡ ¢ η ∂ T + ∂ T − , = −2 + + − + ···

where T ± T ±(x±). = This is the full conformal group in 2 dimensions, generated by T +(x+) and T −(x−). The Virasoro charges are given by 2 Z Q [T ±] dφT ±f . ± = l ±±

D 3 - Asymptotic symmetries =

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This is the full conformal group in 2 dimensions, generated by T +(x+) and T −(x−). The Virasoro charges are given by 2 Z Q [T ±] dφT ±f . ± = l ±±

D 3 - Asymptotic symmetries =

Asymptotic behaviour of These boundary conditions are invariant under diffeomorphisms gravity in three dimensions ηα of the form Marc Henneaux 2 Introduction l 2 η+ T + 2 ∂ T − , Hamiltonian = + 2r − + ··· formulation and surface terms 2 l 2 Asymptotic η− T − 2 ∂ T + , symmetries = + 2r + + ···

Algebra of charges r r ¡ ¢ η ∂ T + ∂ T − , Asymptotically = −2 + + − + ··· anti-de Sitter spaces where T ± T ±(x±). Central charge in = 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The Virasoro charges are given by 2 Z Q [T ±] dφT ±f . ± = l ±±

D 3 - Asymptotic symmetries =

Algebra of charges r r ¡ ¢ η ∂ T + ∂ T − , Asymptotically = −2 + + − + ··· anti-de Sitter spaces where T ± T ±(x±). Central charge in = 3 dimensions This is the full conformal group in 2 dimensions, generated by Chern-Simons reformulation T +(x+) and T −(x−).

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions D 3 - Asymptotic symmetries =

Conclusions The Virasoro charges are given by 2 Z Q [T ±] dφT ±f . ± = l ±±

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Using the transformation rules of the f ’s, 2 l ¡ 3 ¢ δηf 2f ∂ T + T +∂ f ∂ T + ∂ T + ++ = ++ + + + ++ − 2 + + + 2 l ¡ 3 ¢ δηf 2f ∂ T − T −∂ f ∂ T − ∂ T − , −− = −− − + − −− − 2 − + −

and invoking the general theorems, one gets two copies of the Virasoro algebra with central charge 3l c . = 2G

In terms of Fourier modes,

³ c 2 ´ [Ln,Lm] i (n m)Ln m n(n 1)δn m,0 , = − − + + 12 − + ³ c 2 ´ [Len,Lem] i (n m)Len m n(n 1)δn m,0 . = − − + + 12 − +

D 3 - Central charges =

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions and invoking the general theorems, one gets two copies of the Virasoro algebra with central charge 3l c . = 2G

In terms of Fourier modes,

³ c 2 ´ [Ln,Lm] i (n m)Ln m n(n 1)δn m,0 , = − − + + 12 − + ³ c 2 ´ [Len,Lem] i (n m)Len m n(n 1)δn m,0 . = − − + + 12 − +

D 3 - Central charges =

Asymptotic behaviour of Using the transformation rules of the f ’s, gravity in three dimensions l2 Marc Henneaux ¡ 3 ¢ δηf 2f ∂ T + T +∂ f ∂ T + ∂ T + ++ = ++ + + + ++ − 2 + + + Introduction 2 Hamiltonian l ¡ 3 ¢ formulation and δηf 2f ∂ T − T −∂ f ∂ T − ∂ T − , surface terms −− = −− − + − −− − 2 − + −

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions In terms of Fourier modes,

³ c 2 ´ [Ln,Lm] i (n m)Ln m n(n 1)δn m,0 , = − − + + 12 − + ³ c 2 ´ [Len,Lem] i (n m)Len m n(n 1)δn m,0 . = − − + + 12 − +

D 3 - Central charges =

Asymptotic symmetries and invoking the general theorems, one gets two copies of the Algebra of charges Virasoro algebra with central charge Asymptotically anti-de Sitter 3l spaces c . Central charge in = 2G 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions D 3 - Central charges =

Chern-Simons reformulation In terms of Fourier modes,

Conclusions ³ c 2 ´ [Ln,Lm] i (n m)Ln m n(n 1)δn m,0 , = − − + + 12 − + ³ c 2 ´ [Len,Lem] i (n m)Len m n(n 1)δn m,0 . = − − + + 12 − +

Marc Henneaux Asymptotic behaviour of gravity in three dimensions AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Chern-Simons theory. The action reads S[A,A˜] S [A] S [A˜] = CS − CS where A, A˜ are connections taking values in the algebra sl(2,R),

and where SCS[A] is the Chern-Simons action

k Z µ 2 ¶ SCS[A] Tr A dA A A A . = 4π M ∧ + 3 ∧ ∧

The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The action reads S[A,A˜] S [A] S [A˜] = CS − CS where A, A˜ are connections taking values in the algebra sl(2,R),

and where SCS[A] is the Chern-Simons action

k Z µ 2 ¶ SCS[A] Tr A dA A A A . = 4π M ∧ + 3 ∧ ∧

The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotic behaviour of gravity in three dimensions Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory.

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions and where SCS[A] is the Chern-Simons action

k Z µ 2 ¶ SCS[A] Tr A dA A A A . = 4π M ∧ + 3 ∧ ∧

The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotic behaviour of gravity in three dimensions Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory. Hamiltonian The action reads formulation and surface terms S[A,A˜] SCS[A] SCS[A˜] Asymptotic = − symmetries where A, A˜ are connections taking values in the algebra sl(2,R), Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotically and where SCS[A] is the Chern-Simons action anti-de Sitter spaces k Z µ 2 ¶ Central charge in SCS[A] Tr A dA A A A . 3 dimensions = 4π M ∧ + 3 ∧ ∧ Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotically and where SCS[A] is the Chern-Simons action anti-de Sitter spaces k Z µ 2 ¶ Central charge in SCS[A] Tr A dA A A A . 3 dimensions = 4π M ∧ + 3 ∧ ∧ Chern-Simons reformulation Conclusions The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The relationship between the sl(2,R) connections A, A˜ and the gravitational variables (dreibein and spin connection) is

1 1 Aa ωa ea and A˜ a ωa ea, i = i + ` i i = i − ` i

in terms of which one ﬁnds indeed 1 Z µ 1 e ¶ ˜ 3 S[A,A] d x eR 2 = 8πG M 2 + `

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 1 1 Aa ωa ea and A˜ a ωa ea, i = i + ` i i = i − ` i

in terms of which one ﬁnds indeed 1 Z µ 1 e ¶ ˜ 3 S[A,A] d x eR 2 = 8πG M 2 + `

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotic behaviour of gravity in three dimensions Marc Henneaux ˜ Introduction The relationship between the sl(2,R) connections A, A and the

Hamiltonian gravitational variables (dreibein and spin connection) is formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions in terms of which one ﬁnds indeed 1 Z µ 1 e ¶ ˜ 3 S[A,A] d x eR 2 = 8πG M 2 + `

AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotic behaviour of gravity in three dimensions Marc Henneaux ˜ Introduction The relationship between the sl(2,R) connections A, A and the

Hamiltonian gravitational variables (dreibein and spin connection) is formulation and surface terms 1 1 Asymptotic Aa ωa ea and A˜ a ωa ea, symmetries i = i + ` i i = i − ` i Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions AdS gravity and sl(2,R) sl(2,R) Chern-Simons ⊕ theory

Asymptotic behaviour of gravity in three dimensions Marc Henneaux ˜ Introduction The relationship between the sl(2,R) connections A, A and the

Hamiltonian gravitational variables (dreibein and spin connection) is formulation and surface terms 1 1 Asymptotic Aa ωa ea and A˜ a ωa ea, symmetries i = i + ` i i = i − ` i Algebra of charges

Asymptotically anti-de Sitter spaces in terms of which one ﬁnds indeed Central charge in 1 Z µ 1 e ¶ 3 dimensions ˜ 3 S[A,A] d x eR 2 Chern-Simons = 8πG M 2 + ` reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The boundary conditions for asymptotically AdS metrics can be reexpressed in terms of the sl(2,R) connections as

h 1 2π i h 1 X12 i A L(φ,t)X11 rX22 dx+ dr ∼ − r k + − r 2

with the other chirality sector fulﬁlling a similar condition. Here, x± are chiral coordinates, x± t φ and L is an arbitrary function = ± of t and φ. X11,X22,X12 are a Chevalley-Serre basis of sl(2,R). It is convenient to eliminate the leading r-dependence by performing a gauge transformation

1 1 A ∆ Ω∂ Ω− ΩA Ω− . i → i = i + i

Boundary conditions in terms of the connection

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions with the other chirality sector fulﬁlling a similar condition. Here, x± are chiral coordinates, x± t φ and L is an arbitrary function = ± of t and φ. X11,X22,X12 are a Chevalley-Serre basis of sl(2,R). It is convenient to eliminate the leading r-dependence by performing a gauge transformation

1 1 A ∆ Ω∂ Ω− ΩA Ω− . i → i = i + i

Boundary conditions in terms of the connection

Asymptotic behaviour of gravity in three dimensions The boundary conditions for asymptotically AdS metrics can be Marc Henneaux reexpressed in terms of the sl(2,R) connections as Introduction

Hamiltonian h 1 2π i h 1 X12 i formulation and A L(φ,t)X11 rX22 dx+ dr surface terms ∼ − r k + − r 2

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It is convenient to eliminate the leading r-dependence by performing a gauge transformation

1 1 A ∆ Ω∂ Ω− ΩA Ω− . i → i = i + i

Boundary conditions in terms of the connection

Asymptotic behaviour of gravity in three dimensions The boundary conditions for asymptotically AdS metrics can be Marc Henneaux reexpressed in terms of the sl(2,R) connections as Introduction

Hamiltonian h 1 2π i h 1 X12 i formulation and A L(φ,t)X11 rX22 dx+ dr surface terms ∼ − r k + − r 2

Asymptotic symmetries Algebra of charges with the other chirality sector fulﬁlling a similar condition. Here, Asymptotically anti-de Sitter x± are chiral coordinates, x± t φ and L is an arbitrary function spaces = ± of t and φ. X11,X22,X12 are a Chevalley-Serre basis of sl(2,R). Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions in terms of the connection

Asymptotic behaviour of gravity in three dimensions The boundary conditions for asymptotically AdS metrics can be Marc Henneaux reexpressed in terms of the sl(2,R) connections as Introduction

Hamiltonian h 1 2π i h 1 X12 i formulation and A L(φ,t)X11 rX22 dx+ dr surface terms ∼ − r k + − r 2

Conclusions 1 1 A ∆ Ω∂ Ω− ΩA Ω− . i → i = i + i

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This can be achieved. In the new gauge, the connection ∆i has as only non-vanishing components ∆ ∆, given by + ≡ 2π ∆ X22 L(φ,t)X11. ∼ − k

We see that the asymptotic boundary conditions are encoded entirely in the highest-weight component (X11).

Boundary conditions in terms of the connection

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions We see that the asymptotic boundary conditions are encoded entirely in the highest-weight component (X11).

Boundary conditions in terms of the connection

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Boundary conditions in terms of the connection

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian This can be achieved. In the new gauge, the connection ∆i has as formulation and surface terms only non-vanishing components ∆ ∆, given by + ≡ Asymptotic symmetries 2π ∆ X22 L(φ,t)X11. Algebra of charges ∼ − k Asymptotically anti-de Sitter spaces Central charge in We see that the asymptotic boundary conditions are encoded 3 dimensions entirely in the highest-weight component (X ). Chern-Simons 11 reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The boundary conditions are preserved by the residual gauge transformations Λ δ∆ Λ0 [∆,Λ] = + that maintain the behavior at asymptotic inﬁnity.

Here, the prime denotes derivative with respect to x+. Note that Λ does not depend asymptotically on x− in order to preserve ∆ 0 − = at asymptotic inﬁnity, so the derivative with respect to x+ is also the derivative with respect to φ One expands Λ as

Λ εX aX bX . = 22 + 12 + 11

One ﬁnds that the asymptotic behaviour is preserved (δ∆ proportional to X11) if and only if

Asymptotic symmetries

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Here, the prime denotes derivative with respect to x+. Note that Λ does not depend asymptotically on x− in order to preserve ∆ 0 − = at asymptotic inﬁnity, so the derivative with respect to x+ is also the derivative with respect to φ One expands Λ as

Λ εX aX bX . = 22 + 12 + 11

One ﬁnds that the asymptotic behaviour is preserved (δ∆ proportional to X11) if and only if

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One expands Λ as

Λ εX aX bX . = 22 + 12 + 11

One ﬁnds that the asymptotic behaviour is preserved (δ∆ proportional to X11) if and only if

Asymptotic symmetries

Asymptotic behaviour of gravity in three The boundary conditions are preserved by the residual gauge dimensions Marc Henneaux transformations Λ δ∆ Λ0 [∆,Λ] Introduction = + Hamiltonian formulation and that maintain the behavior at asymptotic inﬁnity. surface terms Here, the prime denotes derivative with respect to x . Note that Λ Asymptotic + symmetries does not depend asymptotically on x− in order to preserve ∆ 0 Algebra of charges − = at asymptotic inﬁnity, so the derivative with respect to x+ is also Asymptotically anti-de Sitter the derivative with respect to φ spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One ﬁnds that the asymptotic behaviour is preserved (δ∆ proportional to X11) if and only if

Asymptotic symmetries

Central charge in One expands Λ as 3 dimensions

Chern-Simons Λ εX22 aX12 bX11. reformulation = + + Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotic symmetries

Central charge in One expands Λ as 3 dimensions

Chern-Simons Λ εX22 aX12 bX11. reformulation = + + Conclusions

One ﬁnds that the asymptotic behaviour is preserved (δ∆ proportional to X11) if and only if

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 1 1 2π a ε0, b ε00 εL. = 2 = 2 − k

Furthermore 2π δ∆ δLX11 = − k with

k δL ε000 (εL)0 ε0L. = −4π + +

Asymptotic symmetries

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Furthermore 2π δ∆ δLX11 = − k with

k δL ε000 (εL)0 ε0L. = −4π + +

Asymptotic symmetries

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction 1 1 2π Hamiltonian a ε0, b ε00 εL. formulation and = 2 = 2 − k surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions k δL ε000 (εL)0 ε0L. = −4π + +

Asymptotic symmetries

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction 1 1 2π Hamiltonian a ε0, b ε00 εL. formulation and = 2 = 2 − k surface terms

Asymptotic symmetries Furthermore Algebra of charges 2π δ∆ δLX11 Asymptotically = − k anti-de Sitter spaces with Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotic symmetries

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction 1 1 2π Hamiltonian a ε0, b ε00 εL. formulation and = 2 = 2 − k surface terms

Asymptotic symmetries Furthermore Algebra of charges 2π δ∆ δLX11 Asymptotically = − k anti-de Sitter spaces with Central charge in 3 dimensions

Chern-Simons k δL ε000 (εL)0 ε0L. reformulation = −4π + + Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions On the other hand δ∆ can also be derived by computing its Poisson bracket with the generator of gauge transformations G[Λ]. Following the Regge-Teitelboim method, one ﬁnds that the surface term that must be added to the (bulk) integral of the Gauss constraint is I dφ (εL).

(and thus on-shell G[Λ] H dφ (εL)) =

Virasoro algebra

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Following the Regge-Teitelboim method, one ﬁnds that the surface term that must be added to the (bulk) integral of the Gauss constraint is I dφ (εL).

(and thus on-shell G[Λ] H dφ (εL)) =

Virasoro algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Virasoro algebra

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction On the other hand δ∆ can also be derived by computing its Hamiltonian formulation and Poisson bracket with the generator of gauge transformations surface terms G[Λ]. Asymptotic symmetries Following the Regge-Teitelboim method, one ﬁnds that the Algebra of charges surface term that must be added to the (bulk) integral of the Asymptotically anti-de Sitter Gauss constraint is spaces I

Central charge in dφ (εL). 3 dimensions H Chern-Simons (and thus on-shell G[Λ] dφ (εL)) reformulation = Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Constraints : Fa 0, or more explicitly ij = Fa 0 rφ = . Derivatives in constraint :

∂ ∆a ∂ ∆a 0 r φ − φ r + ··· =

In variation of R Tr ¡ΛF ¢, pick up surface term − rφ I I Tr ¡Λδ∆ ¢ δ Tr ¡Λ∆ ¢ − φ = − φ

Hence, one must add the surface term I ¡ ¢ Tr Λ∆φ .

Parenthesis - Surface terms for 2+1 Chern-Simons theory

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Derivatives in constraint :

∂ ∆a ∂ ∆a 0 r φ − φ r + ··· =

In variation of R Tr ¡ΛF ¢, pick up surface term − rφ I I Tr ¡Λδ∆ ¢ δ Tr ¡Λ∆ ¢ − φ = − φ

Hence, one must add the surface term I ¡ ¢ Tr Λ∆φ .

Parenthesis - Surface terms for 2+1 Chern-Simons theory

Asymptotic a behaviour of Constraints : F 0, or more explicitly gravity in three ij = dimensions a Marc Henneaux F 0 rφ = Introduction . Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions In variation of R Tr ¡ΛF ¢, pick up surface term − rφ I I Tr ¡Λδ∆ ¢ δ Tr ¡Λ∆ ¢ − φ = − φ

Hence, one must add the surface term I ¡ ¢ Tr Λ∆φ .

Parenthesis - Surface terms for 2+1 Chern-Simons theory

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Hence, one must add the surface term I ¡ ¢ Tr Λ∆φ .

Parenthesis - Surface terms for 2+1 Chern-Simons theory

Asymptotic a behaviour of Constraints : F 0, or more explicitly gravity in three ij = dimensions a Marc Henneaux F 0 rφ = Introduction . Hamiltonian formulation and Derivatives in constraint : surface terms Asymptotic ∂ ∆a ∂ ∆a 0 symmetries r φ − φ r + ··· = Algebra of charges Asymptotically R ¡ ¢ anti-de Sitter In variation of Tr ΛFrφ , pick up surface term spaces − I I Central charge in ¡ ¢ ¡ ¢ 3 dimensions Tr Λδ∆φ δ Tr Λ∆φ Chern-Simons − = − reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Parenthesis - Surface terms for 2+1 Chern-Simons theory

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This enables one to read the brackets of the L’s

k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) = −4π φ − + + −

Again, one ﬁnds the Virasoro algebra (with same central charge).

Virasoro algebra

Asymptotic behaviour of gravity in three On the other hand δ∆ can also be derived by computing its dimensions Poisson bracket with the generator of gauge transformations Marc Henneaux G[Λ]. Introduction

Hamiltonian Following the Regge-Teitelboim method, one ﬁnds that the formulation and surface terms surface term that must be added to the (bulk) integral of the Gauss constraint is Asymptotic I symmetries dφ (εL). Algebra of charges Asymptotically H anti-de Sitter (and thus on-shell G[Λ] dφ (εL)) spaces = Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) = −4π φ − + + −

Again, one ﬁnds the Virasoro algebra (with same central charge).

Virasoro algebra

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Again, one ﬁnds the Virasoro algebra (with same central charge).

Virasoro algebra

Chern-Simons reformulation

Conclusions k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) = −4π φ − + + −

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Virasoro algebra

Chern-Simons reformulation

Conclusions k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) = −4π φ − + + −

Again, one ﬁnds the Virasoro algebra (with same central charge).

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. The symmetry algebra is given by two copies of the Virasoro algebra with a non-vanishing central charge that can be expressed in terms of

3l c , = 2G

a fact that has been used to derive the 3D black hole entropy (Strominger 1998). This can be veriﬁed in the standard metric formulation...... or in the Chern-Simons formulation

Conclusions and comments

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The symmetry algebra is given by two copies of the Virasoro algebra with a non-vanishing central charge that can be expressed in terms of

3l c , = 2G

a fact that has been used to derive the 3D black hole entropy (Strominger 1998). This can be veriﬁed in the standard metric formulation...... or in the Chern-Simons formulation

Conclusions and comments

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 3l c , = 2G

a fact that has been used to derive the 3D black hole entropy (Strominger 1998). This can be veriﬁed in the standard metric formulation...... or in the Chern-Simons formulation

Conclusions and comments

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. Introduction

Hamiltonian The symmetry algebra is given by two copies of the Virasoro formulation and surface terms algebra with a non-vanishing central charge that can be Asymptotic expressed in terms of symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions a fact that has been used to derive the 3D black hole entropy (Strominger 1998). This can be veriﬁed in the standard metric formulation...... or in the Chern-Simons formulation

Conclusions and comments

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. Introduction

Algebra of charges 3l Asymptotically c , anti-de Sitter spaces = 2G

Central charge in 3 dimensions

Chern-Simons reformulation

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ... or in the Chern-Simons formulation

Conclusions and comments

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. Introduction

Algebra of charges 3l Asymptotically c , anti-de Sitter spaces = 2G

Central charge in 3 dimensions a fact that has been used to derive the 3D black hole entropy Chern-Simons reformulation (Strominger 1998). Conclusions This can be veriﬁed in the standard metric formulation...

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Conclusions and comments

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Even though locally trivial, three-dimensional gravity with a negative cosmological constant possesses rich asymptotics. Introduction

Algebra of charges 3l Asymptotically c , anti-de Sitter spaces = 2G

Marc Henneaux Asymptotic behaviour of gravity in three dimensions