Chern-Simons reformulation Hypersymmetric black holes in 2+1 gravity AdS gravity can be reformulated as an sl (2, R ) ⊕ sl (2, R ) Marc Henneaux Chern-Simons theory. Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 5 / 28
Chern-Simons reformulation Hypersymmetric black holes in 2+1 gravity AdS gravity can be reformulated as an sl (2, R ) ⊕ sl (2, R ) Marc Henneaux Chern-Simons theory. Introduction The action reads Three- dimensional pure gravity with Λ < 0 I [ A + , A − ] = I CS [ A + ] − I CS [ A − ] Hypergravity Charges and asymptotic where A + , A − are connections taking values in the algebra sl (2, R ), analysis Hypersymmetry bounds Black holes Conclusions 5 / 28
Chern-Simons reformulation Hypersymmetric black holes in 2+1 gravity AdS gravity can be reformulated as an sl (2, R ) ⊕ sl (2, R ) Marc Henneaux Chern-Simons theory. Introduction The action reads Three- dimensional pure gravity with Λ < 0 I [ A + , A − ] = I CS [ A + ] − I CS [ A − ] Hypergravity Charges and asymptotic where A + , A − are connections taking values in the algebra sl (2, R ), analysis Hypersymmetry and where I CS [ A ] is the Chern-Simons action bounds Black holes I CS [ A ] = k � A ∧ dA + 2 � � Conclusions Tr 3 A ∧ A ∧ A . 4 π M 5 / 28
Chern-Simons reformulation Hypersymmetric black holes in 2+1 gravity AdS gravity can be reformulated as an sl (2, R ) ⊕ sl (2, R ) Marc Henneaux Chern-Simons theory. Introduction The action reads Three- dimensional pure gravity with Λ < 0 I [ A + , A − ] = I CS [ A + ] − I CS [ A − ] Hypergravity Charges and asymptotic where A + , A − are connections taking values in the algebra sl (2, R ), analysis Hypersymmetry and where I CS [ A ] is the Chern-Simons action bounds Black holes I CS [ A ] = k � A ∧ dA + 2 � � Conclusions Tr 3 A ∧ A ∧ A . 4 π M The parameter k is related to the (2+1)-dimensional Newton constant G as k = ℓ /4 G , where ℓ is the AdS radius of curvature. 5 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 6 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The relationship between the sl (2, R ) connections A + , A − and the Introduction Three- gravitational variables (dreibein and spin connection) is dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 6 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The relationship between the sl (2, R ) connections A + , A − and the Introduction Three- gravitational variables (dreibein and spin connection) is dimensional pure gravity with Λ < 0 Hypergravity µ + 1 µ − 1 A + a µ = ω a ℓ e a A − a µ = ω a ℓ e a and µ , Charges and µ asymptotic analysis Hypersymmetry bounds Black holes Conclusions 6 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The relationship between the sl (2, R ) connections A + , A − and the Introduction Three- gravitational variables (dreibein and spin connection) is dimensional pure gravity with Λ < 0 Hypergravity µ + 1 µ − 1 A + a µ = ω a ℓ e a A − a µ = ω a ℓ e a and µ , Charges and µ asymptotic analysis Hypersymmetry bounds in terms of which one finds indeed Black holes 1 � � 1 2 eR + e � Conclusions d 3 x I [ e , ω ] = ℓ 2 8 π G M 6 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 7 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The absence of local degrees of freedom manifests itself in the Introduction Chern-Simons formulation through the fact that the connection Three- dimensional pure gravity with Λ < 0 is flat, Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 7 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The absence of local degrees of freedom manifests itself in the Introduction Chern-Simons formulation through the fact that the connection Three- dimensional pure gravity with Λ < 0 is flat, Hypergravity Charges and F = 0, asymptotic analysis Hypersymmetry bounds Black holes Conclusions 7 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The absence of local degrees of freedom manifests itself in the Introduction Chern-Simons formulation through the fact that the connection Three- dimensional pure gravity with Λ < 0 is flat, Hypergravity Charges and F = 0, asymptotic analysis Hypersymmetry bounds which implies that one can locally set it to zero, A = 0, by a gauge Black holes transformation. Conclusions 7 / 28
AdS pure gravity and sl (2, R ) ⊕ sl (2, R ) Chern-Simons theory Hypersymmetric black holes in 2+1 gravity Marc Henneaux The absence of local degrees of freedom manifests itself in the Introduction Chern-Simons formulation through the fact that the connection Three- dimensional pure gravity with Λ < 0 is flat, Hypergravity Charges and F = 0, asymptotic analysis Hypersymmetry bounds which implies that one can locally set it to zero, A = 0, by a gauge Black holes transformation. Conclusions Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate. 7 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 The subalgebra sl (2, R ) is called the “gravitational subalgebra". Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 The subalgebra sl (2, R ) is called the “gravitational subalgebra". Hypergravity (Really, sl (2, R ) ⊕ sl (2, R ) but I will consider explicitly only one Charges and asymptotic sector from now on.) analysis Hypersymmetry bounds Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 The subalgebra sl (2, R ) is called the “gravitational subalgebra". Hypergravity (Really, sl (2, R ) ⊕ sl (2, R ) but I will consider explicitly only one Charges and asymptotic sector from now on.) analysis Hypersymmetry In supergravity, the bosonic subalgebra is the direct sum bounds sl (2, R ) ⊕ G , where G is the “R-symmetry algebra". Black holes Conclusions 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 The subalgebra sl (2, R ) is called the “gravitational subalgebra". Hypergravity (Really, sl (2, R ) ⊕ sl (2, R ) but I will consider explicitly only one Charges and asymptotic sector from now on.) analysis Hypersymmetry In supergravity, the bosonic subalgebra is the direct sum bounds sl (2, R ) ⊕ G , where G is the “R-symmetry algebra". Black holes Conclusions The fermionic generators transform in the 2 of sl (2, R ), which might come with a non-trivial multiplicity (extended supergravities). 8 / 28
D = 3 Pure N -extended Supergravities as Chern-Simons theories Hypersymmetric black holes in 2+1 The Chern-Simons formulation is very convenient because it gravity allows for generalizations. Marc Henneaux Introduction For instance, supergravity is obtained by simply replacing sl (2, R ) Three- by a superalgebra that contains it. dimensional pure gravity with Λ < 0 The subalgebra sl (2, R ) is called the “gravitational subalgebra". Hypergravity (Really, sl (2, R ) ⊕ sl (2, R ) but I will consider explicitly only one Charges and asymptotic sector from now on.) analysis Hypersymmetry In supergravity, the bosonic subalgebra is the direct sum bounds sl (2, R ) ⊕ G , where G is the “R-symmetry algebra". Black holes Conclusions The fermionic generators transform in the 2 of sl (2, R ), which might come with a non-trivial multiplicity (extended supergravities). The first condition ensures that the theory contains gravity and only bosonic fields of “spins" 2 and 1 (and a single “graviton"). The second condition ensures that spinors are spin- 3 2 fields. 8 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 9 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux But one may relax these conditions ! Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 9 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux But one may relax these conditions ! Introduction Three- This leads to higher spin gauge theories in 3D. dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 9 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux But one may relax these conditions ! Introduction Three- This leads to higher spin gauge theories in 3D. dimensional pure gravity with Λ < 0 In 3D, the higher spin gauge theories are simply given by a Hypergravity Chern-Simons theory with appropriate “higher spin" Charges and asymptotic (super)algebra. analysis Hypersymmetry bounds Black holes Conclusions 9 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux But one may relax these conditions ! Introduction Three- This leads to higher spin gauge theories in 3D. dimensional pure gravity with Λ < 0 In 3D, the higher spin gauge theories are simply given by a Hypergravity Chern-Simons theory with appropriate “higher spin" Charges and asymptotic (super)algebra. analysis Hypersymmetry These higher spin (super)algebras are obtained by lifting the bounds above restrictions that limited the spin content to ≤ 2. Black holes Conclusions 9 / 28
Higher spin gauge theories Hypersymmetric black holes in 2+1 gravity Marc Henneaux But one may relax these conditions ! Introduction Three- This leads to higher spin gauge theories in 3D. dimensional pure gravity with Λ < 0 In 3D, the higher spin gauge theories are simply given by a Hypergravity Chern-Simons theory with appropriate “higher spin" Charges and asymptotic (super)algebra. analysis Hypersymmetry These higher spin (super)algebras are obtained by lifting the bounds above restrictions that limited the spin content to ≤ 2. Black holes Conclusions One then considers general (super)algebras containing the gravitational subalgebra sl (2, R ), but with their bosonic subalgebra not necessarily of the form sl (2, R ) ⊕ G . 9 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 10 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity The case of interest to us is obtained by replacing sl (2, R ) by Marc Henneaux osp (1,4). Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 10 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity The case of interest to us is obtained by replacing sl (2, R ) by Marc Henneaux osp (1,4). Introduction More precisely, one replaces the gauge algebra sl (2, R ) ⊕ sl (2, R ) by Three- dimensional pure osp (1 | 4) ⊕ osp (1 | 4), the bosonic subalgebra of which is gravity with Λ < 0 sp (4) ⊕ sp (4). The resulting theory contains automatically gravity Hypergravity since sl (2, R ) ⊂ sp (4). Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 10 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity The case of interest to us is obtained by replacing sl (2, R ) by Marc Henneaux osp (1,4). Introduction More precisely, one replaces the gauge algebra sl (2, R ) ⊕ sl (2, R ) by Three- dimensional pure osp (1 | 4) ⊕ osp (1 | 4), the bosonic subalgebra of which is gravity with Λ < 0 sp (4) ⊕ sp (4). The resulting theory contains automatically gravity Hypergravity since sl (2, R ) ⊂ sp (4). Charges and asymptotic analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry is in contrast with D > 3 where one needs an infinite number of bounds higher spin gauge fields to get a consistent theory. But what is the Black holes spin content ? Conclusions 10 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity The case of interest to us is obtained by replacing sl (2, R ) by Marc Henneaux osp (1,4). Introduction More precisely, one replaces the gauge algebra sl (2, R ) ⊕ sl (2, R ) by Three- dimensional pure osp (1 | 4) ⊕ osp (1 | 4), the bosonic subalgebra of which is gravity with Λ < 0 sp (4) ⊕ sp (4). The resulting theory contains automatically gravity Hypergravity since sl (2, R ) ⊂ sp (4). Charges and asymptotic analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry is in contrast with D > 3 where one needs an infinite number of bounds higher spin gauge fields to get a consistent theory. But what is the Black holes spin content ? Conclusions Assuming principal embedding of sl (2, R ) in sp (4), one gets one spin-2 field, one spin-4 field and one spin- 5 2 field. 10 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity The case of interest to us is obtained by replacing sl (2, R ) by Marc Henneaux osp (1,4). Introduction More precisely, one replaces the gauge algebra sl (2, R ) ⊕ sl (2, R ) by Three- dimensional pure osp (1 | 4) ⊕ osp (1 | 4), the bosonic subalgebra of which is gravity with Λ < 0 sp (4) ⊕ sp (4). The resulting theory contains automatically gravity Hypergravity since sl (2, R ) ⊂ sp (4). Charges and asymptotic analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry is in contrast with D > 3 where one needs an infinite number of bounds higher spin gauge fields to get a consistent theory. But what is the Black holes spin content ? Conclusions Assuming principal embedding of sl (2, R ) in sp (4), one gets one spin-2 field, one spin-4 field and one spin- 5 2 field. The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984). 10 / 28
Some conventions Hypersymmetric Basis of osp (1 | 4) : black holes in 2+1 gravity � � Marc Henneaux � � L i , L j = i − j L i + j , Introduction � � L i , U m = (3 i − m ) U i + m , � 3 Three- � dimensional pure � � L i , S p 2 i − p S i + p , = gravity with Λ < 0 Hypergravity �� �� � � 1 m 2 + n 2 − 2 − 2 m 2 + n 2 − 4 [ U m , U n ] = 2 2 3 ( m − n ) 3 mn − 9 3 ( mn − 6) mn L m + n Charges and asymptotic analysis + 1 � m 2 − mn + n 2 − 7 � 6 ( m − n ) U m + n , Hypersymmetry bounds 1 � 2 m 3 − 8 m 2 p + 20 mp 2 + 82 p − 23 m − 40 p 3 � � � U m , S p = S i + p , Black holes 2 3 3 Conclusions 1 � 6 p 2 − 8 pq + 6 q 2 − 9 � � � S p , S q = U p + q + L p + q . 2 2 3 Here L i , with i = 0, ± 1, stand for the spin-2 generators that span the gravitational sl (2, R ) subalgebra, while U m and S p , with m = 0, ± 1, ± 2, ± 3 and p = ± 1 2 , ± 3 2 , correspond to the spin-4 and fermionic spin- 5 2 generators, respectively. 11 / 28
Dynamics Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 12 / 28
Dynamics Hypersymmetric black holes in 2+1 The action is gravity A + � − I CS [ A − ] � I = I CS Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 12 / 28
Dynamics Hypersymmetric black holes in 2+1 The action is gravity A + � − I CS [ A − ] � I = I CS Marc Henneaux Introduction Three- with dimensional pure gravity with Λ < 0 I CS [ A ] = k 4 � AdA + 2 � � 3 A 3 str . Hypergravity 4 π Charges and asymptotic analysis Here, the level, k 4 = k /10, is expressed in terms of the Newton Hypersymmetry bounds constant and the AdS radius according to k = ℓ /4 G . Black holes Conclusions 12 / 28
Dynamics Hypersymmetric black holes in 2+1 The action is gravity A + � − I CS [ A − ] � I = I CS Marc Henneaux Introduction Three- with dimensional pure gravity with Λ < 0 I CS [ A ] = k 4 � AdA + 2 � � 3 A 3 str . Hypergravity 4 π Charges and asymptotic analysis Here, the level, k 4 = k /10, is expressed in terms of the Newton Hypersymmetry bounds constant and the AdS radius according to k = ℓ /4 G . Black holes str [ ··· ] stands for the supertrace of the fundamental (5 × 5) Conclusions matrix representation. 12 / 28
Dynamics Hypersymmetric black holes in 2+1 The action is gravity A + � − I CS [ A − ] � I = I CS Marc Henneaux Introduction Three- with dimensional pure gravity with Λ < 0 I CS [ A ] = k 4 � AdA + 2 � � 3 A 3 str . Hypergravity 4 π Charges and asymptotic analysis Here, the level, k 4 = k /10, is expressed in terms of the Newton Hypersymmetry bounds constant and the AdS radius according to k = ℓ /4 G . Black holes str [ ··· ] stands for the supertrace of the fundamental (5 × 5) Conclusions matrix representation. The connection reads A + = A i µ U m + ψ p µ L i + B m µ S p and a similar expression holds for A − . 12 / 28
Dynamics Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic where e µ ∼ A + µ − A − µ , analysis Hypersymmetry bounds Black holes Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic where e µ ∼ A + µ − A − µ , analysis µ a ∼ ψ p and the spin- 5 Hypersymmetry 2 field is ψ µ a , γ a ψ µ a = 0 ( ψ α µ ). bounds Black holes Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic where e µ ∼ A + µ − A − µ , analysis µ a ∼ ψ p and the spin- 5 Hypersymmetry 2 field is ψ µ a , γ a ψ µ a = 0 ( ψ α µ ). bounds The action is S [ g µν , h µνρσ , ψ µ a ] = S E + S F + S 5 Black holes 2 + S I Conclusions 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic where e µ ∼ A + µ − A − µ , analysis µ a ∼ ψ p and the spin- 5 Hypersymmetry 2 field is ψ µ a , γ a ψ µ a = 0 ( ψ α µ ). bounds The action is S [ g µν , h µνρσ , ψ µ a ] = S E + S F + S 5 Black holes 2 + S I Conclusions where S E is the Einstein action, S F the (covariantized) Fronsdal 5 2 the (covariantized) spin- 5 action for a spin-4 field, S 2 action and S I stands for the higher order interaction terms necessary to make the theory consistent. 13 / 28
Dynamics Hypersymmetric In terms of the two osp (1 | 4) connections A + and A − , the metric black holes in 2+1 gravity and spin-4 field are defined by Marc Henneaux Introduction Three- dimensional pure g µν ∼ str ( e µ e ν ) , h µνρσ ∼ str ( e µ e ν e ρ e σ ) + astr ( e ( µ e ν ) str ( e ρ e σ ) ), gravity with Λ < 0 Hypergravity Charges and asymptotic where e µ ∼ A + µ − A − µ , analysis µ a ∼ ψ p and the spin- 5 Hypersymmetry 2 field is ψ µ a , γ a ψ µ a = 0 ( ψ α µ ). bounds The action is S [ g µν , h µνρσ , ψ µ a ] = S E + S F + S 5 Black holes 2 + S I Conclusions where S E is the Einstein action, S F the (covariantized) Fronsdal 5 2 the (covariantized) spin- 5 action for a spin-4 field, S 2 action and S I stands for the higher order interaction terms necessary to make the theory consistent. These interaction terms are not known in closed form. They can be constructed perturbatively. 13 / 28
Absence of a well-defined geometry Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 14 / 28
Absence of a well-defined geometry Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction An important (and puzzling !) feature of higher spin gauge Three- dimensional pure gravity with Λ < 0 theories is that the metric g µν transforms under the gauge Hypergravity transformations of the spin-4 gauge field h λµνρ . Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 14 / 28
Absence of a well-defined geometry Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction An important (and puzzling !) feature of higher spin gauge Three- dimensional pure gravity with Λ < 0 theories is that the metric g µν transforms under the gauge Hypergravity transformations of the spin-4 gauge field h λµνρ . Charges and There is no known definition of a geometry that would be asymptotic analysis invariant under higher spin gauge symmetries. Hypersymmetry bounds Black holes Conclusions 14 / 28
Absence of a well-defined geometry Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction An important (and puzzling !) feature of higher spin gauge Three- dimensional pure gravity with Λ < 0 theories is that the metric g µν transforms under the gauge Hypergravity transformations of the spin-4 gauge field h λµνρ . Charges and There is no known definition of a geometry that would be asymptotic analysis invariant under higher spin gauge symmetries. Hypersymmetry bounds In particular, given a solution to the field equation, there is no Black holes known way to ascribe to it a well-defined causal structure. Conclusions 14 / 28
Absence of a well-defined geometry Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction An important (and puzzling !) feature of higher spin gauge Three- dimensional pure gravity with Λ < 0 theories is that the metric g µν transforms under the gauge Hypergravity transformations of the spin-4 gauge field h λµνρ . Charges and There is no known definition of a geometry that would be asymptotic analysis invariant under higher spin gauge symmetries. Hypersymmetry bounds In particular, given a solution to the field equation, there is no Black holes known way to ascribe to it a well-defined causal structure. Conclusions We shall come back to that question later. 14 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 15 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity The boundary conditions were first investigated in the metric Marc Henneaux formulation and a precise definition of what is meant by Introduction “asymptotically anti-de Sitter metric" was given. Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 15 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity The boundary conditions were first investigated in the metric Marc Henneaux formulation and a precise definition of what is meant by Introduction “asymptotically anti-de Sitter metric" was given. Three- dimensional pure These boundary conditions can be reformulated in terms of the gravity with Λ < 0 Chern-Simons connection. Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 15 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity The boundary conditions were first investigated in the metric Marc Henneaux formulation and a precise definition of what is meant by Introduction “asymptotically anti-de Sitter metric" was given. Three- dimensional pure These boundary conditions can be reformulated in terms of the gravity with Λ < 0 Chern-Simons connection. Hypergravity Charges and It turns out that (in a suitable gauge) they take exactly the same asymptotic analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry conditions, namely bounds Black holes Conclusions 15 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity The boundary conditions were first investigated in the metric Marc Henneaux formulation and a precise definition of what is meant by Introduction “asymptotically anti-de Sitter metric" was given. Three- dimensional pure These boundary conditions can be reformulated in terms of the gravity with Λ < 0 Chern-Simons connection. Hypergravity Charges and It turns out that (in a suitable gauge) they take exactly the same asymptotic analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry conditions, namely bounds Black holes r →∞ L ± 1 − 2 π � 1 � Conclusions A ± k L ± � � � � r , ϕ − → ϕ L ∓ 1 + O , ϕ r and � 1 � A ± r − r →∞ O → . r 15 / 28
Boundary conditions - Pure gravity Hypersymmetric We first consider pure gravity black holes in 2+1 gravity The boundary conditions were first investigated in the metric Marc Henneaux formulation and a precise definition of what is meant by Introduction “asymptotically anti-de Sitter metric" was given. Three- dimensional pure These boundary conditions can be reformulated in terms of the gravity with Λ < 0 Chern-Simons connection. Hypergravity Charges and It turns out that (in a suitable gauge) they take exactly the same asymptotic analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry conditions, namely bounds Black holes r →∞ L ± 1 − 2 π � 1 � Conclusions A ± k L ± � � � � r , ϕ − → ϕ L ∓ 1 + O , ϕ r and � 1 � A ± r − r →∞ O → . r Coussaert, Henneaux, van Driel 1995 15 / 28
Asymptotic symmetries Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 16 / 28
Asymptotic symmetries Hypersymmetric black holes in 2+1 gravity Marc Henneaux The “asymptotic symmetries" are those gauge transformations Introduction i = ∂ i Λ ± + [ A ± δ A ± i , Λ ± ] that preserve the boundary conditions, i.e., Three- dimensional pure such that A ± i + δ A ± i fulfills also the boundary conditions. gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 16 / 28
Asymptotic symmetries Hypersymmetric black holes in 2+1 gravity Marc Henneaux The “asymptotic symmetries" are those gauge transformations Introduction i = ∂ i Λ ± + [ A ± δ A ± i , Λ ± ] that preserve the boundary conditions, i.e., Three- dimensional pure such that A ± i + δ A ± i fulfills also the boundary conditions. gravity with Λ < 0 Hypergravity They are given by Charges and asymptotic �� L ± 1 − 2 π � analysis Λ ± k L ± � � � ± ǫ ± ϕ ϕ L ∓ 1 − → Hypersymmetry r →∞ bounds L 0 ± 1 Black holes ∓ ǫ ′ 2 ǫ ′′ � � � � ϕ ϕ L ∓ 1 ± ± Conclusions 16 / 28
Asymptotic symmetries Hypersymmetric black holes in 2+1 gravity Marc Henneaux The “asymptotic symmetries" are those gauge transformations Introduction i = ∂ i Λ ± + [ A ± δ A ± i , Λ ± ] that preserve the boundary conditions, i.e., Three- dimensional pure such that A ± i + δ A ± i fulfills also the boundary conditions. gravity with Λ < 0 Hypergravity They are given by Charges and asymptotic �� L ± 1 − 2 π � analysis Λ ± k L ± � � � ± ǫ ± ϕ ϕ L ∓ 1 − → Hypersymmetry r →∞ bounds L 0 ± 1 Black holes ∓ ǫ ′ 2 ǫ ′′ � � � � ϕ ϕ L ∓ 1 ± ± Conclusions � � The functions ǫ ± ϕ are arbitrary functions of ϕ and parametrize the asymptotic symmetries. 16 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity These generators obey the Virasoro algebra. Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity These generators obey the Virasoro algebra. Charges and asymptotic More precisely, the Fourier components L ± n obey, in terms of the analysis Poisson bracket, the Virasoro algebra with the classical central Hypersymmetry bounds charge c = 6 k = 3 ℓ /2 G , Black holes Conclusions 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity These generators obey the Virasoro algebra. Charges and asymptotic More precisely, the Fourier components L ± n obey, in terms of the analysis Poisson bracket, the Virasoro algebra with the classical central Hypersymmetry bounds charge c = 6 k = 3 ℓ /2 G , Black holes Conclusions m + n + k L ± m , L ± PB = ( m − n ) L ± 2 m 3 δ m + n ,0 . � � i n 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity These generators obey the Virasoro algebra. Charges and asymptotic More precisely, the Fourier components L ± n obey, in terms of the analysis Poisson bracket, the Virasoro algebra with the classical central Hypersymmetry bounds charge c = 6 k = 3 ℓ /2 G , Black holes Conclusions m + n + k L ± m , L ± PB = ( m − n ) L ± 2 m 3 δ m + n ,0 . � � i n � L + m , L − � and commute between themselves PB = 0 (2 D n conformal algebra). 17 / 28
Generators of asymptotic symmetries - Virasoro algebra Hypersymmetric Furthermore, one easily finds that the generators of the black holes in 2+1 gravity asymptotic symmetry algebra are given explictly by the L ± ’s Marc Henneaux themselves (when the constraints hold) and read explicitly Introduction � L ± � Three- � � � Q ± [ ǫ ± ] = ± ǫ ± ϕ ϕ d ϕ dimensional pure gravity with Λ < 0 r →∞ Hypergravity These generators obey the Virasoro algebra. Charges and asymptotic More precisely, the Fourier components L ± n obey, in terms of the analysis Poisson bracket, the Virasoro algebra with the classical central Hypersymmetry bounds charge c = 6 k = 3 ℓ /2 G , Black holes Conclusions m + n + k L ± m , L ± PB = ( m − n ) L ± 2 m 3 δ m + n ,0 . � � i n � L + m , L − � and commute between themselves PB = 0 (2 D n conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions. 17 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 18 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction The same asymptotic analysis can be performed for hypergravity. Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 18 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction The same asymptotic analysis can be performed for hypergravity. Three- dimensional pure One gets an enhancement of the asymptotic algebra, from the gravity with Λ < 0 Virasoro algebra to the W (2, 5 2 ,4) -superalgebra, which contains the Hypergravity Charges and Virasoro generators but also generators of higher conformal asymptotic analysis weights. Hypersymmetry bounds Black holes Conclusions 18 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction The same asymptotic analysis can be performed for hypergravity. Three- dimensional pure One gets an enhancement of the asymptotic algebra, from the gravity with Λ < 0 Virasoro algebra to the W (2, 5 2 ,4) -superalgebra, which contains the Hypergravity Charges and Virasoro generators but also generators of higher conformal asymptotic analysis weights. Hypersymmetry How does this proceed ? bounds Black holes Conclusions 18 / 28
Hypergravity Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction The same asymptotic analysis can be performed for hypergravity. Three- dimensional pure One gets an enhancement of the asymptotic algebra, from the gravity with Λ < 0 Virasoro algebra to the W (2, 5 2 ,4) -superalgebra, which contains the Hypergravity Charges and Virasoro generators but also generators of higher conformal asymptotic analysis weights. Hypersymmetry How does this proceed ? bounds Black holes Sugra : Henneaux, Maoz, Schwimmer (2000) ; Conclusions Higher spins : S.-J. Rey + MH (2010) ; A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen (2010) 18 / 28
Boundary conditions Hypersymmetric black holes in 2+1 gravity Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 19 / 28
Boundary conditions Hypersymmetric The asymptotic conditions that generalize those found for pure black holes in 2+1 gravity gravity are again of Drinfeld-Sokolov type Marc Henneaux Introduction Three- dimensional pure gravity with Λ < 0 Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 19 / 28
Boundary conditions Hypersymmetric The asymptotic conditions that generalize those found for pure black holes in 2+1 gravity gravity are again of Drinfeld-Sokolov type Marc Henneaux Introduction Three- r →∞ L ± 1 − 2 π L ∓ 1 + π U ∓ 3 − 2 π A ± k L ± � 5 k U ± � k ψ ± � � � � � � dimensional pure r , ϕ ϕ ϕ ϕ − → S ∓ 3 2 , gravity with Λ < 0 ϕ Hypergravity Charges and asymptotic analysis Hypersymmetry bounds Black holes Conclusions 19 / 28
Boundary conditions Hypersymmetric The asymptotic conditions that generalize those found for pure black holes in 2+1 gravity gravity are again of Drinfeld-Sokolov type Marc Henneaux Introduction Three- r →∞ L ± 1 − 2 π L ∓ 1 + π U ∓ 3 − 2 π A ± k L ± � 5 k U ± � k ψ ± � � � � � � dimensional pure r , ϕ ϕ ϕ ϕ − → S ∓ 3 2 , gravity with Λ < 0 ϕ Hypergravity Charges and Again, the non trivial fields L ± � � , U ± � � and ψ ± ( ϕ ) appear asymptotic ϕ ϕ analysis along the lowest (highest)-weight generators. Hypersymmetry bounds Black holes Conclusions 19 / 28
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