Asymptotic Behaviour of Gravity in Three Dimensions Marc Henneaux

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and Asymptotic behaviour of gravity in three surface terms Asymptotic dimensions symmetries

Algebra of charges

Asymptotically anti-de Sitter Marc Henneaux spaces

Central charge in 3 dimensions

Chern-Simons reformulation IPMU, Tokyo, February 2011 Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Three-dimensional gravity, with or without cosmological constant, is at ﬁrst sight trivial. Indeed, the Riemann tensor is completely determined by the Ricci tensor, and the Einstein equations imply therefore that the solutions are spaces of constant curvature,

R 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Indeed, the Riemann tensor is completely determined by the Ricci tensor, and the Einstein equations imply therefore that the solutions are spaces of constant curvature,

Introduction

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions However, the case with negative cosmological constant has attracted in the last years a lot of interest.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux Three-dimensional gravity, with or without cosmological Introduction constant, is at ﬁrst sight trivial. Hamiltonian formulation and Indeed, the Riemann tensor is completely determined by the surface terms Ricci tensor, and the Einstein equations imply therefore that the Asymptotic symmetries solutions are spaces of constant curvature, Algebra of charges

Asymptotically ¡ ¢ anti-de Sitter 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

Supergravity in 3 dimensions

Higher Spin Theory

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 Supergravity in 3 dimensions attracted in the last years a lot of interest. Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This is because even though the theory is locally trivial, it is not globally so. It admits black hole solutions which are obtained through identiﬁcations of anti-de-Sitter space. It has an interesting behaviour at inﬁnity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . = 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It admits black hole solutions which are obtained through identiﬁcations of anti-de-Sitter space. It has an interesting behaviour at inﬁnity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . = 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction This is because even though the theory is locally trivial, it is not

Hamiltonian globally so. formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It has an interesting behaviour at inﬁnity, where the asymptotic symmetry algebra is given by two copies of the Virasoro algebra with central charge c 3` . = 2G Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction This is because even though the theory is locally trivial, it is not

Hamiltonian globally so. formulation and surface terms It admits black hole solutions which are obtained through Asymptotic symmetries identiﬁcations of anti-de-Sitter space.

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Three-dimensional gravity with negative cosmological constant presents therefore many of the interesting features of the higher-dimensional versions, but in a much simpler context.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction This is because even though the theory is locally trivial, it is not

Hamiltonian globally so. formulation and surface terms It admits black hole solutions which are obtained through Asymptotic symmetries identiﬁcations of anti-de-Sitter space. Algebra of charges It has an interesting behaviour at inﬁnity, where the asymptotic Asymptotically symmetry algebra is given by two copies of the Virasoro algebra anti-de Sitter spaces 3` with central charge c 2G . Central charge in = 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction This is because even though the theory is locally trivial, it is not

Supergravity in 3 higher-dimensional versions, but in a much simpler context. dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The purpose of the talk is to explain the asymptotic structure of three-dimensional anti-de Sitter gravity. To that end, it is necessary to review the Hamiltonian approach to conserved charges in gravitation theory and understand the possible occurence of central charges. We shall ﬁrst consider pure gravity. We shall then consider supersymmetric and higher spin extensions of 3D AdS gravity, using the Chern-Simons formulation.

Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions To that end, it is necessary to review the Hamiltonian approach to conserved charges in gravitation theory and understand the possible occurence of central charges. We shall ﬁrst consider pure gravity. We shall then consider supersymmetric and higher spin 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

Supergravity in 3 dimensions

Higher Spin Theory

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 review the Hamiltonian approach symmetries to conserved charges in gravitation theory and understand the Algebra of charges possible occurence of central charges. Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Introduction

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction 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. Supergravity in 3 dimensions

Higher Spin Theory

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. It does not assume that the constraints hold (“off-shell method"). 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

Supergravity in 3 dimensions

Higher Spin Theory

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. It does not assume that the constraints hold (“off-shell method"). 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 symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It has 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. It does not assume that the constraints hold (“off-shell method"). 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

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It can be applied to asymptotic symmetries that are not exact symmetries. It does not assume that the constraints hold (“off-shell method"). 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 Deﬁning conserved charges (energy, angular momentum, ...) in Introduction general relativity is a subtle issue. Hamiltonian formulation and One approach is based on the Hamiltonian formulation of surface terms gravity. Asymptotic symmetries It has the advantage of clearly connecting the charges to the Algebra of charges generators of the corresponding symmetries. Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions It does not assume that the constraints hold (“off-shell method"). 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

Central charge in symmetries. 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

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

Central charge in symmetries. 3 dimensions

Chern-Simons It does not assume that the constraints hold (“off-shell method"). reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Central charge in symmetries. 3 dimensions

Chern-Simons It does not assume that the constraints hold (“off-shell method"). reformulation Finally, it gives direct access to the algebra of the charges and the Supergravity in 3 dimensions possibility of non-Poissonian actions (“central charges"). Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The dynamical evolution of the ﬁelds expresses how they change as one goes from one spacelike hypersurface to the next. n

ξ = ∂/∂t ⊥ ξ n .(t+dt, xk)

Σ . k (t, x ) ξk ∂/∂xk

ξ = ξ⊥ n + ξk ∂/∂xk

Gravitational Hamiltonian

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

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions n

ξ = ∂/∂t ⊥ ξ n .(t+dt, xk)

Σ . k (t, x ) ξk ∂/∂xk

ξ = ξ⊥ n + ξk ∂/∂xk

Gravitational Hamiltonian

Asymptotic behaviour of IDEA : conserved charges = generators of the corresponding gravity in three dimensions asymptotic symmetries in the Poisson bracket Marc Henneaux The dynamical evolution of the ﬁelds expresses how they change Introduction as one goes from one spacelike hypersurface to the next. Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Chern-Simons reformulation

Supergravity in 3 dimensions Σ . k (t, x ) ξk ∂/∂xk Higher Spin Theory Conclusions ξ = ξ⊥ n + ξk ∂/∂xk

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" Q[ξ] 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)

The bulk term vanishes on account of the constraints

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

The bulk term vanishes on account of the constraints

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

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

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions R with “bulk term" ddx(ξ H ξkH ) = Σ ⊥ + k and “surface term" Q[ξ] H ( local expression) . = ∂Σ The bulk term vanishes on account of the constraints

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions and “surface term" Q[ξ] H ( local expression) . = ∂Σ The bulk term vanishes on account of the constraints

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Hamiltonian formulation and k ∂ surface terms Under an arbitrary surface deformation ξ ξ⊥n ξ of Σ, the = + ∂xk Asymptotic functional F changes as (Dirac, ADM) symmetries δ F [F,H[ξ]] Algebra of charges ξ = Asymptotically where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) anti-de Sitter = R + with “bulk term" ddx(ξ H ξkH ) spaces = Σ ⊥ + k Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

H 0, H 0. ≈ k ≈

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Hamiltonian formulation and k ∂ surface terms Under an arbitrary surface deformation ξ ξ⊥n ξ of Σ, the = + ∂xk Asymptotic functional F changes as (Dirac, ADM) symmetries δ F [F,H[ξ]] Algebra of charges ξ = Asymptotically where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) anti-de Sitter = R d + k spaces with “bulk term" d x(ξ⊥H ξ Hk) = Σ H + Central charge in and “surface term" Q[ξ] ∂Σ( local expression) . 3 dimensions =

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions ij Let F[gij(x),π (x), ] be a functional deﬁned on a spacelike Marc Henneaux ··· hypersurface Σ. Introduction

Hamiltonian formulation and k ∂ surface terms Under an arbitrary surface deformation ξ ξ⊥n ξ of Σ, the = + ∂xk Asymptotic functional F changes as (Dirac, ADM) symmetries δ F [F,H[ξ]] Algebra of charges ξ = Asymptotically where H[ξ] “bulk term" “surface term" (Regge-Teitelboim) anti-de Sitter = R d + k spaces with “bulk term" d x(ξ⊥H ξ Hk) = Σ H + Central charge in and “surface term" Q[ξ] ∂Σ( local expression) . 3 dimensions = The bulk term vanishes on account of the constraints Chern-Simons reformulation

Supergravity in 3 H 0, Hk 0. dimensions ≈ ≈ Higher Spin Theory

Conclusions

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., δH[ξ] “Volume term" = 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

(no δφ,k, δφ,km etc)

Gravitational Hamiltonian

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

Introduction volume term ?

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions i.e., δH[ξ] “Volume term" = 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

(no δφ,k, δφ,km etc)

Gravitational Hamiltonian

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

Introduction volume term ?

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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

(no δφ,k, δφ,km etc)

Gravitational Hamiltonian

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

Introduction volume term ?

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions (no δφ,k, δφ,km etc)

Gravitational Hamiltonian

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

Introduction volume term ?

Hamiltonian formulation and Requirement : H[ξ] should have well-defined functional surface terms derivatives in the class of fields under consideration (“be Asymptotic symmetries differentiable") Algebra of charges i.e., δH[ξ] “Volume term" Asymptotically = anti-de Sitter spaces where the volume term contains only undifferentiated variations Central charge in Z 3 dimensions d ³ ij ij contributions from´ d x A (x)δgij(x) Bij(x)δπ (x) Chern-Simons + + other fields reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

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

Introduction volume term ?

Supergravity in 3 dimensions (no δφ,k, δφ,km etc)

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Now, one has in general δH[ξ] “Volume term" M[ξ] δQ[ξ] = + + where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form. One must therefore impose

M[ξ] δQ[ξ] 0. + =

(Regge and Teitelboim 1976)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions where M[ξ] is the surface term arising from integrations by parts to bring the volume term in desired form. One must therefore impose

M[ξ] δQ[ξ] 0. + =

(Regge and Teitelboim 1976)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Now, one has in general δH[ξ] “Volume term" M[ξ] δQ[ξ] Hamiltonian = + + formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One must therefore impose

M[ξ] δQ[ξ] 0. + =

(Regge and Teitelboim 1976)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Now, one has in general δH[ξ] “Volume term" M[ξ] δQ[ξ] Hamiltonian = + + formulation and where M[ ] is the surface term arising from integrations by parts surface terms ξ

Asymptotic to bring the volume term in desired form. symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions (Regge and Teitelboim 1976)

Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Now, one has in general δH[ξ] “Volume term" M[ξ] δQ[ξ] Hamiltonian = + + formulation and where M[ ] is the surface term arising from integrations by parts surface terms ξ

Asymptotic to bring the volume term in desired form. symmetries One must therefore impose Algebra of charges

Asymptotically anti-de Sitter M[ξ] δQ[ξ] 0. spaces + = Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction Now, one has in general δH[ξ] “Volume term" M[ξ] δQ[ξ] Hamiltonian = + + formulation and where M[ ] is the surface term arising from integrations by parts surface terms ξ

Asymptotic to bring the volume term in desired form. symmetries One must therefore impose Algebra of charges

Asymptotically anti-de Sitter M[ξ] δQ[ξ] 0. spaces + = Central charge in 3 dimensions

Chern-Simons reformulation (Regge and Teitelboim 1976)

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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] The condition M[ξ] δQ[ξ] 0 implies that the variation δH[ξ] + = reduces to the volume term.

Gravitational Hamiltonian

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

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Gravitational Hamiltonian

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

Introduction Z µ ∂ ∂ ¶ Hamiltonian d H H formulation and δH[ξ] d x ξ⊥ δgij ξ⊥ δgij,m δQ[ξ] surface terms = ∂gij + ∂gij,m + ··· +

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Gravitational Hamiltonian

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

Introduction Z µ ∂ ∂ ¶ Hamiltonian d H H formulation and δH[ξ] d x ξ⊥ δgij ξ⊥ δgij,m δQ[ξ] surface terms = ∂gij + ∂gij,m + ··· + Asymptotic "Ã ! # symmetries Z µ ¶ d ∂H ∂H Algebra of charges d x ξ⊥ ξ⊥ δgij = ∂gij − ∂gij,m + ··· + ··· Asymptotically ,m anti-de Sitter Z µ ¶ spaces d 1 ∂H Central charge in d − Sm ξ⊥ δgij δQ[ξ] 3 dimensions + ∂gij,m + ··· +

Chern-Simons | {z } reformulation M[ξ]

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Gravitational Hamiltonian

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

Chern-Simons | {z } reformulation M[ξ] Supergravity in 3 The condition M[ξ] δQ[ξ] 0 implies that the variation δH[ξ] dimensions + = Higher Spin reduces to the volume term. Theory

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

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 Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Higher Spin If one modiﬁes the Lagrangian (by the topological mass term in 3 Theory dimensions, or by matter ﬁelds, or in any other (local) way), the Conclusions 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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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. Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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. 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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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. 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. Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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.

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions The asymptotic symmetries with zero charges are the true gauge

Higher Spin symmetries. Theory

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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...). Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

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

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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 Evaluated on the background, this relation reduces to Chern-Simons reformulation Supergravity in 3 δBQA KAB. dimensions = Higher Spin Theory If the background is invariant (strictly and not just Conclusions 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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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. Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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 − Supergravity in 3 dimensions

Higher Spin Theory

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

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The full metric is ³ ´ ds2 g0 h dxµdxν, = µν + µν

0 where gµν is the anti-de Sitter metric and where the deviation hµν goes to zero at inﬁnity. What determines the precise fall-off (“boundary conditions") ?

Asymptotically anti-de Sitter spaces - 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 0 where gµν is the anti-de Sitter metric and where the deviation hµν goes to zero at inﬁnity. What determines the precise fall-off (“boundary conditions") ?

Asymptotically anti-de Sitter spaces - Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian The full metric is formulation and surface terms 2 ³ 0 ´ µ ν Asymptotic ds gµν hµν dx dx , symmetries = +

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions What determines the precise fall-off (“boundary conditions") ?

Asymptotically anti-de Sitter spaces - Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian The full metric is formulation and surface terms 2 ³ 0 ´ µ ν Asymptotic ds gµν hµν dx dx , symmetries = +

Algebra of charges

Asymptotically 0 anti-de Sitter where g is the anti-de Sitter metric and where the deviation h spaces µν µν

Central charge in goes to zero at inﬁnity. 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotically anti-de Sitter spaces - Boundary conditions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian The full metric is formulation and surface terms 2 ³ 0 ´ µ ν Asymptotic ds gµν hµν dx dx , symmetries = +

Algebra of charges

Asymptotically 0 anti-de Sitter where g is the anti-de Sitter metric and where the deviation h spaces µν µν

Central charge in goes to zero at inﬁnity. 3 dimensions What determines the precise fall-off (“boundary conditions") ? Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Higher Spin Theory

Conclusions

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. Supergravity in 3 dimensions They have been extended to include “slowly decaying" scalar Higher Spin Theory fields in M.H. + C. Martínez + R. Troncoso + J. Zanelli (2004, 2007).

Conclusions

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).

Asymptotic Symmetry Group (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

Supergravity in 3 dimensions

Higher Spin Theory

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 ∞

Asymptotic Symmetry Group (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

Supergravity in 3 dimensions

Higher Spin Theory

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).

Asymptotic Symmetry Group (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

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Asymptotic Symmetry Group (D 4) ≥

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Asymptotic Symmetry Group (D 4) ≥

Higher Spin One recovers in particular the Abbott-Deser energy (1982). Theory

Conclusions

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) One finds that for D 3, the asymptotic symmetry group is much = larger than O(2,2) and is instead the conformal group C (2) in 2 dimensions, which is infinite-dimensional and parametrized by two arbitrary functions T +(x+) and T −(x−), x± t φ. = ±

D 3 - Asymptotic Symmetry Group =

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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) One finds that for D 3, the asymptotic symmetry group is much = larger than O(2,2) and is instead the conformal group C (2) in 2 dimensions, which is infinite-dimensional and parametrized by two arbitrary functions T +(x+) and T −(x−), x± t φ. = ±

D 3 - Asymptotic Symmetry Group =

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction The D 3 case was investigated in M.H.+ David Brown (1986) for = Hamiltonian pure gravity or gravity coupled to “fastly decaying" matter ﬁelds. formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One ﬁnds that for D 3, the asymptotic symmetry group is much = larger than O(2,2) and is instead the conformal group C (2) in 2 dimensions, which is inﬁnite-dimensional and parametrized by two arbitrary functions T +(x+) and T −(x−), x± t φ. = ±

D 3 - Asymptotic Symmetry Group =

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction The D 3 case was investigated in M.H.+ David Brown (1986) for = Hamiltonian pure gravity or gravity coupled to “fastly decaying" matter ﬁelds. formulation and surface terms The analysis has been extended to include “slowly decaying" Asymptotic symmetries scalar ﬁelds in M.H. + C. Martínez + R. Troncoso + J. Zanelli

Algebra of charges (2002), and to include a topological mass term in Asymptotically MH-C.Martínez-R.Troncoso 2009) anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Algebra of charges (2002), and to include a topological mass term in Asymptotically MH-C.Martínez-R.Troncoso 2009) anti-de Sitter spaces One ﬁnds that for D 3, the asymptotic symmetry group is much Central charge in = (2) 3 dimensions larger than O(2,2) and is instead the conformal group C in 2

Chern-Simons dimensions, which is inﬁnite-dimensional and parametrized by reformulation two arbitrary functions T +(x+) and T −(x−), x± t φ. Supergravity in 3 = ± dimensions

Higher Spin Theory

Conclusions

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

Background Asymptotic isometries symmetries D 4 O(D 1,2) O(D 1,2) ≥ − − D 3 O(2,2) C (2) O(2,2) = ⊃ For d 3 the asymptotic symmetry group C (2) cannot be realized = as the isometry group of some background. Central charges are allowed and do, in fact, occur. This was shown in M.H.+ David Brown (1986) using the metric formulation and the above Hamiltonian techniques. The central charge turns out to be equal to

3` c . = 2G

Summary

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Background Asymptotic isometries symmetries D 4 O(D 1,2) O(D 1,2) ≥ − − D 3 O(2,2) C (2) O(2,2) = ⊃ For d 3 the asymptotic symmetry group C (2) cannot be realized = as the isometry group of some background. Central charges are allowed and do, in fact, occur. This was shown in M.H.+ David Brown (1986) using the metric formulation and the above Hamiltonian techniques. The central charge turns out to be equal to

3` c . = 2G

Summary

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

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions For d 3 the asymptotic symmetry group C (2) cannot be realized = as the isometry group of some background. Central charges are allowed and do, in fact, occur. This was shown in M.H.+ David Brown (1986) using the metric formulation and the above Hamiltonian techniques. The central charge turns out to be equal to

3` c . = 2G

Summary

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This was shown in M.H.+ David Brown (1986) using the metric formulation and the above Hamiltonian techniques. The central charge turns out to be equal to

3` c . = 2G

Summary

Asymptotic behaviour of The situation is thus the following : gravity in three dimensions Marc Henneaux Background Asymptotic Introduction isometries symmetries Hamiltonian formulation and D 4 O(D 1,2) O(D 1,2) surface terms ≥ − (2) − Asymptotic D 3 O(2,2) C O(2,2) symmetries = ⊃ Algebra of charges For d 3 the asymptotic symmetry group C (2) cannot be realized Asymptotically = anti-de Sitter as the isometry group of some background. Central charges are spaces allowed and do, in fact, occur. Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Summary

Supergravity in 3 charge turns out to be equal to dimensions

Higher Spin 3` Theory c . = 2G Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions The parameter k is related to the (2+1)-dimensional Newton

Higher Spin constant G as k `/4G, where ` is the AdS radius of curvature. Theory =

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Some conventions

Asymptotic behaviour of gravity in three dimensions Marc Henneaux µ 0 1 ¶ Introduction X − , 11 = 0 0 Hamiltonian formulation and surface terms µ 1 0 ¶ Asymptotic X12 − , symmetries = 0 1 Algebra of charges µ ¶ Asymptotically 0 0 anti-de Sitter X22 . spaces = 1 0

Central charge in 3 dimensions [X12,X11] 2X11,[X11,X22] X12,[X12,X22] 2X22. Chern-Simons = − = = reformulation

Supergravity in 3 dimensions

Higher Spin (Chevalley-Serre basis, up to some obvious signs) Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 1 1 dimensions Ai ∆i Ω∂iΩ− ΩAiΩ− . Higher Spin → = + Theory

Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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 = + + Supergravity in 3 dimensions

Higher Spin Theory

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 = + + Supergravity in 3 dimensions

Higher Spin Theory One ﬁnds that the asymptotic behaviour is preserved Conclusions (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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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 + + Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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 = Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Supergravity in 3 dimensions Hence, one must add the surface term

Higher Spin I Theory ¡ ¢ Tr Λ∆φ . Conclusions

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Supergravity in 3 k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) dimensions = −4π φ − + + − Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions Virasoro algebra

Chern-Simons reformulation

Supergravity in 3 k 3 ¡ ¢ [L(φ),L(φ0)] ∂ δ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) dimensions = −4π φ − + + − Higher Spin Theory

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

Marc Henneaux Asymptotic behaviour of gravity in three dimensions On can generalize easily the analysis to supergravity because it is also a Chern-Simons theory (Achúcarro and Townsend 1986). The gauge superalgebra contains sl(2,R) as a subalgebra of its bosonic part (in each chiral sector). The analysis was done in MH, Maoz and Schwimmer 2000, where the appropriate boundary conditions where given. One ﬁnds asymptotically two copies of the (extended) super-Virasoro algebras...... with nonlinearities in the currents in the extended cases (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The gauge superalgebra contains sl(2,R) as a subalgebra of its bosonic part (in each chiral sector). The analysis was done in MH, Maoz and Schwimmer 2000, where the appropriate boundary conditions where given. One ﬁnds asymptotically two copies of the (extended) super-Virasoro algebras...... with nonlinearities in the currents in the extended cases (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Hamiltonian also a Chern-Simons theory (Achúcarro and Townsend 1986). formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The analysis was done in MH, Maoz and Schwimmer 2000, where the appropriate boundary conditions where given. One ﬁnds asymptotically two copies of the (extended) super-Virasoro algebras...... with nonlinearities in the currents in the extended cases (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Hamiltonian also a Chern-Simons theory (Achúcarro and Townsend 1986). formulation and surface terms The gauge superalgebra contains sl(2,R) as a subalgebra of its Asymptotic bosonic part (in each chiral sector). symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions One ﬁnds asymptotically two copies of the (extended) super-Virasoro algebras...... with nonlinearities in the currents in the extended cases (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Algebra of charges The analysis was done in MH, Maoz and Schwimmer 2000, where

Asymptotically the appropriate boundary conditions where given. anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ...with nonlinearities in the currents in the extended cases (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Algebra of charges The analysis was done in MH, Maoz and Schwimmer 2000, where

Asymptotically the appropriate boundary conditions where given. anti-de Sitter spaces One ﬁnds asymptotically two copies of the (extended) Central charge in 3 dimensions super-Virasoro algebras...

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky).

(Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Algebra of charges The analysis was done in MH, Maoz and Schwimmer 2000, where

Asymptotically the appropriate boundary conditions where given. anti-de Sitter spaces One ﬁnds asymptotically two copies of the (extended) Central charge in 3 dimensions super-Virasoro algebras... Chern-Simons ...with nonlinearities in the currents in the extended cases reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions (Extended) super-Virasoro algebras

Asymptotic behaviour of gravity in three dimensions Marc Henneaux On can generalize easily the analysis to supergravity because it is Introduction

Algebra of charges The analysis was done in MH, Maoz and Schwimmer 2000, where

Supergravity in 3 (algebras of Knizhnik, Bershadsky, Fradkin and Linetsky). dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions On can also consider higher spin theories, which are again Chern-Simons theories in 3 dimensions (Blencowe 1989). This was done in MH and Rey 2010. The gauge algebra is the direct sum hs(1,1) hs(1,1) of two copies ⊕ of the higher spin algebra hs(1,1) deﬁned as follows. hs(1,1) is the algebra of polynomials of even degree in two commuting variables ξ1, ξ2 with no constant term, equipped with the Lie bracket

³ ∂ ∂ ∂ ∂ ´ ¯ [f ,g] sin f (η)g(ζ)¯ . = ∂η1 ∂ζ2 − ∂η2 ∂ζ1 ¯η ζ ξ = =

The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions This was done in MH and Rey 2010. The gauge algebra is the direct sum hs(1,1) hs(1,1) of two copies ⊕ of the higher spin algebra hs(1,1) deﬁned as follows. hs(1,1) is the algebra of polynomials of even degree in two commuting variables ξ1, ξ2 with no constant term, equipped with the Lie bracket

³ ∂ ∂ ∂ ∂ ´ ¯ [f ,g] sin f (η)g(ζ)¯ . = ∂η1 ∂ζ2 − ∂η2 ∂ζ1 ¯η ζ ξ = =

The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W algebra ∞

Asymptotic behaviour of gravity in three dimensions On can also consider higher spin theories, which are again Marc Henneaux Chern-Simons theories in 3 dimensions (Blencowe 1989). Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The gauge algebra is the direct sum hs(1,1) hs(1,1) of two copies ⊕ of the higher spin algebra hs(1,1) deﬁned as follows. hs(1,1) is the algebra of polynomials of even degree in two commuting variables ξ1, ξ2 with no constant term, equipped with the Lie bracket

³ ∂ ∂ ∂ ∂ ´ ¯ [f ,g] sin f (η)g(ζ)¯ . = ∂η1 ∂ζ2 − ∂η2 ∂ζ1 ¯η ζ ξ = =

The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W algebra ∞

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions hs(1,1) is the algebra of polynomials of even degree in two commuting variables ξ1, ξ2 with no constant term, equipped with the Lie bracket

³ ∂ ∂ ∂ ∂ ´ ¯ [f ,g] sin f (η)g(ζ)¯ . = ∂η1 ∂ζ2 − ∂η2 ∂ζ1 ¯η ζ ξ = =

The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W algebra ∞

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ³ ∂ ∂ ∂ ∂ ´ ¯ [f ,g] sin f (η)g(ζ)¯ . = ∂η1 ∂ζ2 − ∂η2 ∂ζ1 ¯η ζ ξ = =

The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W algebra ∞

Asymptotic behaviour of gravity in three dimensions On can also consider higher spin theories, which are again Marc Henneaux Chern-Simons theories in 3 dimensions (Blencowe 1989). Introduction This was done in MH and Rey 2010. Hamiltonian formulation and The gauge algebra is the direct sum hs(1,1) hs(1,1) of two copies surface terms ⊕ Asymptotic of the higher spin algebra hs(1,1) deﬁned as follows. symmetries hs(1,1) is the algebra of polynomials of even degree in two Algebra of charges 1 2 Asymptotically commuting variables ξ , ξ with no constant term, equipped anti-de Sitter spaces with the Lie bracket

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The polynomials of degree 2 form a subalgebra isomorphic to sl(2,R).

W algebra ∞

Central charge in 3 dimensions ³ ∂ ∂ ∂ ∂ ´ ¯ Chern-Simons [f ,g] sin f ( )g( )¯ . 1 2 2 1 η ζ ¯ reformulation = ∂η ∂ζ − ∂η ∂ζ η ζ ξ = = Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions W algebra ∞

Central charge in 3 dimensions ³ ∂ ∂ ∂ ∂ ´ ¯ Chern-Simons [f ,g] sin f ( )g( )¯ . 1 2 2 1 η ζ ¯ reformulation = ∂η ∂ζ − ∂η ∂ζ η ζ ξ = = Supergravity in 3 dimensions

Higher Spin The polynomials of degree 2 form a subalgebra isomorphic to Theory sl(2,R). Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The correspondence is :

1 1 2 1 2 1 2 2 X11 (ξ ) , X12 ξ ξ , X22 (ξ ) . = 2 = = 2

The subalgebra hs(1,1) splits into a direct sum of representations of sl(2,R): hs(1,1) k 1Dk, = ⊕ ≥ where the spin k representation Dk corresponds to the homogeneous polynomials of degree 2k. The trivial representation D0 does not appear because we consider traceless polynomials. The action is the Chern-Simons action and contains the Einstein action + action for a ﬁeld of spin 3 + action for a ﬁeld of spin 4 + ...

W 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 1 1 2 1 2 1 2 2 X11 (ξ ) , X12 ξ ξ , X22 (ξ ) . = 2 = = 2

W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The correspondence is :

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The subalgebra hs(1,1) splits into a direct sum of representations of sl(2,R): hs(1,1) k 1Dk, = ⊕ ≥ where the spin k representation Dk corresponds to the homogeneous polynomials of degree 2k. The trivial representation D0 does not appear because we consider traceless polynomials. The action is the Chern-Simons action and contains the Einstein action + action for a ﬁeld of spin 3 + action for a ﬁeld of spin 4 + ...

W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The correspondence is :

Introduction 1 1 2 1 2 1 2 2 Hamiltonian X (ξ ) , X ξ ξ , X (ξ ) . formulation and 11 12 22 surface terms = 2 = = 2

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions where the spin k representation Dk corresponds to the homogeneous polynomials of degree 2k. The trivial representation D0 does not appear because we consider traceless polynomials. The action is the Chern-Simons action and contains the Einstein action + action for a ﬁeld of spin 3 + action for a ﬁeld of spin 4 + ...

W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The correspondence is :

Introduction 1 1 2 1 2 1 2 2 Hamiltonian X (ξ ) , X ξ ξ , X (ξ ) . formulation and 11 12 22 surface terms = 2 = = 2

Asymptotic symmetries The subalgebra hs(1,1) splits into a direct sum of representations Algebra of charges of sl(2,R): hs(1,1) D , Asymptotically k 1 k anti-de Sitter = ⊕ ≥ spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The action is the Chern-Simons action and contains the Einstein action + action for a ﬁeld of spin 3 + action for a ﬁeld of spin 4 + ...

W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The correspondence is :

Introduction 1 1 2 1 2 1 2 2 Hamiltonian X (ξ ) , X ξ ξ , X (ξ ) . formulation and 11 12 22 surface terms = 2 = = 2

Asymptotic symmetries The subalgebra hs(1,1) splits into a direct sum of representations Algebra of charges of sl(2,R): hs(1,1) D , Asymptotically k 1 k anti-de Sitter = ⊕ ≥ spaces where the spin k representation Dk corresponds to the Central charge in homogeneous polynomials of degree 2k. The trivial 3 dimensions representation D does not appear because we consider traceless Chern-Simons 0 reformulation polynomials. Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux The correspondence is :

Introduction 1 1 2 1 2 1 2 2 Hamiltonian X (ξ ) , X ξ ξ , X (ξ ) . formulation and 11 12 22 surface terms = 2 = = 2

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure gravity

2π ∆ X22 L(φ,t)X11 ∼ − k

are :

X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) ∼ + k 1 ∼ ≥

where N(2k,0) are arbitrary functions of t and φ (with N 2π L(φ,t)) (2,0) = − k The asymptotic symmetry algebra is now W , with Mj 1 N(2j,0) ∞ + ∼ being the generators with conformal weight j 1. +

W 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 2π ∆ X22 L(φ,t)X11 ∼ − k

are :

X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) ∼ + k 1 ∼ ≥

W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian formulation and surface terms

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions are :

X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) ∼ + k 1 ∼ ≥

W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian 2π formulation and ∆ X22 L(φ,t)X11 surface terms ∼ − k

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) ∼ + k 1 ∼ ≥

W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian 2π formulation and ∆ X22 L(φ,t)X11 surface terms ∼ − k

Asymptotic symmetries

Algebra of charges are :

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions where N(2k,0) are arbitrary functions of t and φ (with N 2π L(φ,t)) (2,0) = − k The asymptotic symmetry algebra is now W , with Mj 1 N(2j,0) ∞ + ∼ being the generators with conformal weight j 1. +

W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian 2π formulation and ∆ X22 L(φ,t)X11 surface terms ∼ − k

Asymptotic symmetries

Algebra of charges are :

Asymptotically anti-de Sitter spaces X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) Central charge in ∼ + k 1 ∼ 3 dimensions ≥

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The asymptotic symmetry algebra is now W , with Mj 1 N(2j,0) ∞ + ∼ being the generators with conformal weight j 1. +

W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian 2π formulation and ∆ X22 L(φ,t)X11 surface terms ∼ − k

Asymptotic symmetries

Algebra of charges are :

Asymptotically anti-de Sitter spaces X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) Central charge in ∼ + k 1 ∼ 3 dimensions ≥

Chern-Simons reformulation where N(2k,0) are arbitrary functions of t and φ (with Supergravity in 3 N 2π L(φ,t)) dimensions (2,0) = − k Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions W algebra ∞

Asymptotic behaviour of gravity in three dimensions The asymptotic conditions that generalize those found for pure Marc Henneaux gravity

Introduction

Hamiltonian 2π formulation and ∆ X22 L(φ,t)X11 surface terms ∼ − k

Asymptotic symmetries

Algebra of charges are :

Asymptotically anti-de Sitter spaces X (2k,0) (2k,0) 1 2k ∆ X22 N(2k,0)X , X (ξ ) Central charge in ∼ + k 1 ∼ 3 dimensions ≥

Chern-Simons reformulation where N(2k,0) are arbitrary functions of t and φ (with Supergravity in 3 N 2π L(φ,t)) dimensions (2,0) = − k Higher Spin The asymptotic symmetry algebra is now W , with Mj 1 N(2j,0) Theory ∞ + ∼ being the generators with conformal weight j 1. Conclusions +

Marc Henneaux Asymptotic behaviour of gravity in three dimensions W algebra ∞

Asymptotic behaviour of gravity in three dimensions Marc Henneaux In particular, if one restricts one’s attention to L and M M , ≡ 3 Introduction

Hamiltonian k 3 ¡ ¢ formulation and [L(φ),L(φ0)] ∂φδ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) surface terms = −4π − + + − ¡ ¢ Asymptotic [L(φ),M(φ0)] M(φ) 2M(φ0) ∂φδ(φ φ0) symmetries = + − Algebra of charges 1 k 5 5 ¡ ¢ 3 [M(φ),M(φ0)] ∂φδ(φ φ0) L(φ) L(φ0) ∂φδ(φ φ0) Asymptotically = 288 · 2π − − 144 + − anti-de Sitter spaces 1 ¡ ¢ L00(φ) L00(φ0) ∂φδ(φ φ0) Central charge in + 48 + − 3 dimensions 1 2π ¡ 2 2 ¢ Chern-Simons L (φ) L (φ0) ∂φδ(φ φ0). reformulation + 9 · k + − Supergravity in 3 dimensions (non-linear W3-algebra) Higher Spin Theory

Conclusions

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. The Chern-Simons formulation enables one to straightforwardly extend the analysis to supergravity or HSAdS3.

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

Supergravity in 3 dimensions

Higher Spin Theory

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

Conclusions and comments

Asymptotic behaviour of gravity in three Even though locally trivial, three-dimensional gravity with a dimensions Marc Henneaux 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

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

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

Conclusions and comments

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

Introduction The symmetry algebra is given by two copies of the Virasoro

Hamiltonian algebra with a non-vanishing central charge that can be formulation and surface terms expressed in terms of

Asymptotic symmetries

Algebra of charges

Asymptotically anti-de Sitter spaces

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions 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. The Chern-Simons formulation enables one to straightforwardly extend the analysis to supergravity or HSAdS3.

Conclusions and comments

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

Introduction The symmetry algebra is given by two copies of the Virasoro

Hamiltonian algebra with a non-vanishing central charge that can be formulation and surface terms expressed in terms of

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

Central charge in 3 dimensions

Chern-Simons reformulation

Supergravity in 3 dimensions

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions ... or in the Chern-Simons formulation. The Chern-Simons formulation enables one to straightforwardly extend the analysis to supergravity or HSAdS3.

Conclusions and comments

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

Introduction The symmetry algebra is given by two copies of the Virasoro

Hamiltonian algebra with a non-vanishing central charge that can be formulation and surface terms expressed in terms of

Higher Spin Theory

Conclusions

Marc Henneaux Asymptotic behaviour of gravity in three dimensions The Chern-Simons formulation enables one to straightforwardly extend the analysis to supergravity or HSAdS3.

Conclusions and comments

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

Introduction The symmetry algebra is given by two copies of the Virasoro

Hamiltonian algebra with a non-vanishing central charge that can be formulation and surface terms expressed in terms of

Asymptotic symmetries 3l Algebra of charges c , Asymptotically = 2G anti-de Sitter spaces Central charge in a fact that has been used to derive the 3D black hole entropy 3 dimensions (Strominger 1998). Chern-Simons reformulation This can be veriﬁed in the standard metric formulation... Supergravity in 3 dimensions ... or in the Chern-Simons formulation. Higher Spin Theory

Conclusions

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

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

Introduction The symmetry algebra is given by two copies of the Virasoro

Hamiltonian algebra with a non-vanishing central charge that can be formulation and surface terms expressed in terms of

Marc Henneaux Asymptotic behaviour of gravity in three dimensions