Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity Hypersymmetric black holes in 2+1 gravity
Charges and asymptotic analysis
Hypersymmetry Marc Henneaux bounds
Black holes
Conclusions ETH, November 2015
1 / 28 Supersymmetry is well known to have deep consequences. In the context of higher spin gauge theories, supersymmetry is generalized to include fermionic symmetries described by parameters of spin 3/2, 5/2 etc Do these symmetries also have interesting implications ? In particular, do they imply “hypersymmetry" bounds ? The answer turns out to be affirmative, as can be seen for instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
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
2 / 28 In the context of higher spin gauge theories, supersymmetry is generalized to include fermionic symmetries described by parameters of spin 3/2, 5/2 etc Do these symmetries also have interesting implications ? In particular, do they imply “hypersymmetry" bounds ? The answer turns out to be affirmative, as can be seen for instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
2 / 28 Do these symmetries also have interesting implications ? In particular, do they imply “hypersymmetry" bounds ? The answer turns out to be affirmative, as can be seen for instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure In the context of higher spin gauge theories, supersymmetry is gravity with Λ 0 < generalized to include fermionic symmetries described by Hypergravity parameters of spin 3/2, 5/2 etc Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
2 / 28 In particular, do they imply “hypersymmetry" bounds ? The answer turns out to be affirmative, as can be seen for instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure In the context of higher spin gauge theories, supersymmetry is gravity with Λ 0 < generalized to include fermionic symmetries described by Hypergravity parameters of spin 3/2, 5/2 etc Charges and asymptotic analysis Do these symmetries also have interesting implications ?
Hypersymmetry bounds
Black holes
Conclusions
2 / 28 The answer turns out to be affirmative, as can be seen for instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure In the context of higher spin gauge theories, supersymmetry is gravity with Λ 0 < generalized to include fermionic symmetries described by Hypergravity parameters of spin 3/2, 5/2 etc Charges and asymptotic analysis Do these symmetries also have interesting implications ?
Hypersymmetry bounds In particular, do they imply “hypersymmetry" bounds ?
Black holes
Conclusions
2 / 28 Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure In the context of higher spin gauge theories, supersymmetry is gravity with Λ 0 < generalized to include fermionic symmetries described by Hypergravity parameters of spin 3/2, 5/2 etc Charges and asymptotic analysis Do these symmetries also have interesting implications ?
Hypersymmetry bounds In particular, do they imply “hypersymmetry" bounds ? Black holes The answer turns out to be affirmative, as can be seen for Conclusions instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions.
2 / 28 Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction Supersymmetry is well known to have deep consequences. Three- dimensional pure In the context of higher spin gauge theories, supersymmetry is gravity with Λ 0 < generalized to include fermionic symmetries described by Hypergravity parameters of spin 3/2, 5/2 etc Charges and asymptotic analysis Do these symmetries also have interesting implications ?
Hypersymmetry bounds In particular, do they imply “hypersymmetry" bounds ? Black holes The answer turns out to be affirmative, as can be seen for Conclusions instance by analysing anti-de Sitter hypergravity in 2 1 + dimensions. Based on joint work with A. Pérez, D. Tempo, R. Troncoso (2015)
2 / 28 Anti-de Sitter Einstein gravity in three spacetime dimensions Anti-de Sitter Hypergravity in three spacetime dimensions Charges and asymptotic analysis Hypersymmetry bounds and black holes Conclusions
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
3 / 28 Anti-de Sitter Hypergravity in three spacetime dimensions Charges and asymptotic analysis Hypersymmetry bounds and black holes Conclusions
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity Anti-de Sitter Einstein gravity in three spacetime dimensions
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
3 / 28 Charges and asymptotic analysis Hypersymmetry bounds and black holes Conclusions
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity Anti-de Sitter Einstein gravity in three spacetime dimensions
Charges and asymptotic Anti-de Sitter Hypergravity in three spacetime dimensions analysis
Hypersymmetry bounds
Black holes
Conclusions
3 / 28 Hypersymmetry bounds and black holes Conclusions
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity Anti-de Sitter Einstein gravity in three spacetime dimensions
Charges and asymptotic Anti-de Sitter Hypergravity in three spacetime dimensions analysis
Hypersymmetry Charges and asymptotic analysis bounds
Black holes
Conclusions
3 / 28 Conclusions
Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity Anti-de Sitter Einstein gravity in three spacetime dimensions
Charges and asymptotic Anti-de Sitter Hypergravity in three spacetime dimensions analysis
Hypersymmetry Charges and asymptotic analysis bounds Hypersymmetry bounds and black holes Black holes
Conclusions
3 / 28 Introduction
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure I will successively discuss : gravity with Λ 0 < Hypergravity Anti-de Sitter Einstein gravity in three spacetime dimensions
Charges and asymptotic Anti-de Sitter Hypergravity in three spacetime dimensions analysis
Hypersymmetry Charges and asymptotic analysis bounds Hypersymmetry bounds and black holes Black holes
Conclusions Conclusions
3 / 28 The AdS algebra in D dimensions is so(D 1,2) − In three dimensions, this gives so(2,2). But so(2,2) is isomorphic to so(2,1) so(2,1) ⊕ and so(2,1) sl(2,R) ' so that so(2,2) is isomorphic to sl(2,R) sl(2,R). ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
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
4 / 28 In three dimensions, this gives so(2,2). But so(2,2) is isomorphic to so(2,1) so(2,1) ⊕ and so(2,1) sl(2,R) ' so that so(2,2) is isomorphic to sl(2,R) sl(2,R). ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
4 / 28 But so(2,2) is isomorphic to so(2,1) so(2,1) ⊕ and so(2,1) sl(2,R) ' so that so(2,2) is isomorphic to sl(2,R) sl(2,R). ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity In three dimensions, this gives so(2,2).
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
4 / 28 and so(2,1) sl(2,R) ' so that so(2,2) is isomorphic to sl(2,R) sl(2,R). ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity In three dimensions, this gives so(2,2). Charges and But so(2,2) is isomorphic to so(2,1) so(2,1) asymptotic ⊕ analysis
Hypersymmetry bounds
Black holes
Conclusions
4 / 28 so that so(2,2) is isomorphic to sl(2,R) sl(2,R). ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity In three dimensions, this gives so(2,2). Charges and But so(2,2) is isomorphic to so(2,1) so(2,1) asymptotic ⊕ analysis and so(2,1) sl(2,R) Hypersymmetry ' bounds
Black holes
Conclusions
4 / 28 Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity In three dimensions, this gives so(2,2). Charges and But so(2,2) is isomorphic to so(2,1) so(2,1) asymptotic ⊕ analysis and so(2,1) sl(2,R) Hypersymmetry ' bounds so that so(2,2) is isomorphic to sl(2,R) sl(2,R). Black holes ⊕
Conclusions
4 / 28 Anti-de Sitter group in three dimensions
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The AdS algebra in D dimensions is so(D 1,2) gravity with Λ 0 − < Hypergravity In three dimensions, this gives so(2,2). Charges and But so(2,2) is isomorphic to so(2,1) so(2,1) asymptotic ⊕ analysis and so(2,1) sl(2,R) Hypersymmetry ' bounds so that so(2,2) is isomorphic to sl(2,R) sl(2,R). Black holes ⊕ Note : one has also sl(2,R) sp(2,R) su(1,1) and thus the chain Conclusions ' ' of isomorphisms so(2,1) sl(2,R) sp(2,R) su(1,1) ' ' '
4 / 28 AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Chern-Simons theory. The action reads
I[A+,A−] I [A+] I [A−] = CS − CS
where A+, A− are connections taking values in the algebra sl(2,R),
and where ICS[A] is the Chern-Simons action
k Z µ 2 ¶ ICS[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. =
Chern-Simons reformulation
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
5 / 28 The action reads
I[A+,A−] I [A+] I [A−] = CS − CS
where A+, A− are connections taking values in the algebra sl(2,R),
and where ICS[A] is the Chern-Simons action
k Z µ 2 ¶ ICS[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. =
Chern-Simons reformulation
Hypersymmetric black holes in 2+1 gravity Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory.
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
5 / 28 and where ICS[A] is the Chern-Simons action
k Z µ 2 ¶ ICS[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. =
Chern-Simons reformulation
Hypersymmetric black holes in 2+1 gravity Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory.
Three- dimensional pure The action reads gravity with Λ 0 < Hypergravity I[A+,A−] ICS[A+] ICS[A−] Charges and = − asymptotic analysis where A+, A− are connections taking values in the algebra sl(2,R), Hypersymmetry bounds
Black holes
Conclusions
5 / 28 The parameter k is related to the (2+1)-dimensional Newton constant G as k `/4G, where ` is the AdS radius of curvature. =
Chern-Simons reformulation
Hypersymmetric black holes in 2+1 gravity Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory.
Three- dimensional pure The action reads gravity with Λ 0 < Hypergravity I[A+,A−] ICS[A+] ICS[A−] Charges and = − asymptotic analysis where A+, A− are connections taking values in the algebra sl(2,R), Hypersymmetry bounds and where ICS[A] is the Chern-Simons action Black holes Z µ ¶ Conclusions k 2 ICS[A] Tr A dA A A A . = 4π M ∧ + 3 ∧ ∧
5 / 28 Chern-Simons reformulation
Hypersymmetric black holes in 2+1 gravity Marc Henneaux AdS gravity can be reformulated as an sl(2,R) sl(2,R) ⊕ Introduction Chern-Simons theory.
Three- dimensional pure The action reads gravity with Λ 0 < Hypergravity I[A+,A−] ICS[A+] ICS[A−] Charges and = − asymptotic analysis where A+, A− are connections taking values in the algebra sl(2,R), Hypersymmetry bounds and where ICS[A] is the Chern-Simons action Black holes Z µ ¶ Conclusions k 2 ICS[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. =
5 / 28 The relationship between the sl(2,R) connections A+, A− and the gravitational variables (dreibein and spin connection) is
a a 1 a a a 1 a A+ ω e and A− ω e , µ = µ + ` µ µ = µ − ` µ
in terms of which one finds indeed Z µ ¶ 1 3 1 e I[e,ω] d x eR 2 = 8πG M 2 + `
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 a a 1 a a a 1 a A+ ω e and A− ω e , µ = µ + ` µ µ = µ − ` µ
in terms of which one finds indeed Z µ ¶ 1 3 1 e I[e,ω] d x eR 2 = 8πG M 2 + `
AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The relationship between the sl(2,R) connections A+, A− and the Three- dimensional pure gravitational variables (dreibein and spin connection) is gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
6 / 28 in terms of which one finds indeed Z µ ¶ 1 3 1 e I[e,ω] d x eR 2 = 8πG M 2 + `
AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The relationship between the sl(2,R) connections A+, A− and the Three- dimensional pure gravitational variables (dreibein and spin connection) is gravity with Λ 0 < Hypergravity a a 1 a a a 1 a Charges and Aµ+ ωµ eµ and Aµ− ωµ eµ, 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
Introduction The relationship between the sl(2,R) connections A+, A− and the Three- dimensional pure gravitational variables (dreibein and spin connection) is gravity with Λ 0 < Hypergravity a a 1 a a a 1 a Charges and Aµ+ ωµ eµ and Aµ− ωµ eµ, asymptotic = + ` = − ` analysis
Hypersymmetry bounds in terms of which one finds indeed Black holes Z µ ¶ Conclusions 1 3 1 e I[e,ω] d x eR 2 = 8πG M 2 + `
6 / 28 The absence of local degrees of freedom manifests itself in the Chern-Simons formulation through the fact that the connection is flat,
F 0, =
which implies that one can locally set it to zero, A 0, by a gauge = transformation. Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate.
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 F 0, =
which implies that one can locally set it to zero, A 0, by a gauge = transformation. Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate.
AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The absence of local degrees of freedom manifests itself in the Three- dimensional pure Chern-Simons formulation through the fact that the connection gravity with Λ 0 < is flat, Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
7 / 28 which implies that one can locally set it to zero, A 0, by a gauge = transformation. Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate.
AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The absence of local degrees of freedom manifests itself in the Three- dimensional pure Chern-Simons formulation through the fact that the connection gravity with Λ 0 < is flat, Hypergravity
Charges and asymptotic F 0, analysis = Hypersymmetry bounds
Black holes
Conclusions
7 / 28 Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate.
AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The absence of local degrees of freedom manifests itself in the Three- dimensional pure Chern-Simons formulation through the fact that the connection gravity with Λ 0 < is flat, Hypergravity
Charges and asymptotic F 0, analysis = Hypersymmetry bounds Black holes which implies that one can locally set it to zero, A 0, by a gauge = Conclusions transformation.
7 / 28 AdS pure gravity and sl(2,R) sl(2,R) ⊕ Chern-Simons theory
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The absence of local degrees of freedom manifests itself in the Three- dimensional pure Chern-Simons formulation through the fact that the connection gravity with Λ 0 < is flat, Hypergravity
Charges and asymptotic F 0, analysis = Hypersymmetry bounds Black holes which implies that one can locally set it to zero, A 0, by a gauge = Conclusions transformation. Note that the Chern-Simons gauge transformations enable one to go to gauges where the triad is degenerate.
7 / 28 The Chern-Simons formulation is very convenient because it allows for generalizations. For instance, supergravity is obtained by simply replacing sl(2,R) by a superalgebra that contains it. The subalgebra sl(2,R) is called the “gravitational subalgebra". (Really, sl(2,R) sl(2,R) but I will consider explicitly only one ⊕ sector from now on.) In supergravity, the bosonic subalgebra is the direct sum sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
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 For instance, supergravity is obtained by simply replacing sl(2,R) by a superalgebra that contains it. The subalgebra sl(2,R) is called the “gravitational subalgebra". (Really, sl(2,R) sl(2,R) but I will consider explicitly only one ⊕ sector from now on.) In supergravity, the bosonic subalgebra is the direct sum sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations.
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
8 / 28 The subalgebra sl(2,R) is called the “gravitational subalgebra". (Really, sl(2,R) sl(2,R) but I will consider explicitly only one ⊕ sector from now on.) In supergravity, the bosonic subalgebra is the direct sum sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
8 / 28 (Really, sl(2,R) sl(2,R) but I will consider explicitly only one ⊕ sector from now on.) In supergravity, the bosonic subalgebra is the direct sum sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. 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 In supergravity, the bosonic subalgebra is the direct sum sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. gravity with Λ 0 < The subalgebra sl(2,R) is called the “gravitational subalgebra". Hypergravity
Charges and (Really, sl(2,R) sl(2,R) but I will consider explicitly only one asymptotic ⊕ analysis sector from now on.)
Hypersymmetry bounds
Black holes
Conclusions
8 / 28 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"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. gravity with Λ 0 < The subalgebra sl(2,R) is called the “gravitational subalgebra". Hypergravity
Charges and (Really, sl(2,R) sl(2,R) but I will consider explicitly only one asymptotic ⊕ analysis sector from now on.)
Hypersymmetry bounds In supergravity, the bosonic subalgebra is the direct sum
Black holes sl(2,R) G , where G is the “R-symmetry algebra". ⊕ Conclusions
8 / 28 The first condition ensures that the theory contains gravity and only bosonic fields of “spins" 2 and 1 (and a single “graviton"). 3 The second condition ensures that spinors are spin- 2 fields.
D 3 Pure N-extended Supergravities as = Chern-Simons theories
Hypersymmetric black holes in 2+1 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. gravity with Λ 0 < The subalgebra sl(2,R) is called the “gravitational subalgebra". Hypergravity
Charges and (Really, sl(2,R) sl(2,R) but I will consider explicitly only one asymptotic ⊕ analysis sector from now on.)
Hypersymmetry bounds In supergravity, the bosonic subalgebra is the direct sum
Black holes sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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 gravity The Chern-Simons formulation is very convenient because it Marc Henneaux allows for generalizations. Introduction For instance, supergravity is obtained by simply replacing sl(2,R) Three- dimensional pure by a superalgebra that contains it. gravity with Λ 0 < The subalgebra sl(2,R) is called the “gravitational subalgebra". Hypergravity
Charges and (Really, sl(2,R) sl(2,R) but I will consider explicitly only one asymptotic ⊕ analysis sector from now on.)
Hypersymmetry bounds In supergravity, the bosonic subalgebra is the direct sum
Black holes sl(2,R) G , where G is the “R-symmetry algebra". ⊕ 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"). 3 The second condition ensures that spinors are spin- 2 fields.
8 / 28 But one may relax these conditions ! This leads to higher spin gauge theories in 3D. In 3D, the higher spin gauge theories are simply given by a Chern-Simons theory with appropriate “higher spin" (super)algebra. These higher spin (super)algebras are obtained by lifting the above restrictions that limited the spin content to 2. ≤ 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 . ⊕
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 This leads to higher spin gauge theories in 3D. In 3D, the higher spin gauge theories are simply given by a Chern-Simons theory with appropriate “higher spin" (super)algebra. These higher spin (super)algebras are obtained by lifting the above restrictions that limited the spin content to 2. ≤ 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 . ⊕
Higher spin gauge theories
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction But one may relax these conditions ! Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
9 / 28 In 3D, the higher spin gauge theories are simply given by a Chern-Simons theory with appropriate “higher spin" (super)algebra. These higher spin (super)algebras are obtained by lifting the above restrictions that limited the spin content to 2. ≤ 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 . ⊕
Higher spin gauge theories
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction But one may relax these conditions ! Three- dimensional pure This leads to higher spin gauge theories in 3D. gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
9 / 28 These higher spin (super)algebras are obtained by lifting the above restrictions that limited the spin content to 2. ≤ 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 . ⊕
Higher spin gauge theories
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction But one may relax these conditions ! Three- dimensional pure This leads to higher spin gauge theories in 3D. gravity with Λ 0 < Hypergravity In 3D, the higher spin gauge theories are simply given by a
Charges and Chern-Simons theory with appropriate “higher spin" asymptotic analysis (super)algebra.
Hypersymmetry bounds
Black holes
Conclusions
9 / 28 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 . ⊕
Higher spin gauge theories
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction But one may relax these conditions ! Three- dimensional pure This leads to higher spin gauge theories in 3D. gravity with Λ 0 < Hypergravity In 3D, the higher spin gauge theories are simply given by a
Charges and Chern-Simons theory with appropriate “higher spin" asymptotic analysis (super)algebra.
Hypersymmetry bounds These higher spin (super)algebras are obtained by lifting the
Black holes above restrictions that limited the spin content to 2. ≤ Conclusions
9 / 28 Higher spin gauge theories
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction But one may relax these conditions ! Three- dimensional pure This leads to higher spin gauge theories in 3D. gravity with Λ 0 < Hypergravity In 3D, the higher spin gauge theories are simply given by a
Charges and Chern-Simons theory with appropriate “higher spin" asymptotic analysis (super)algebra.
Hypersymmetry bounds These higher spin (super)algebras are obtained by lifting the
Black holes above restrictions that limited the spin content to 2. ≤ 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 The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by ⊕ osp(1 4) osp(1 4), the bosonic subalgebra of which is | ⊕ | sp(4) sp(4). The resulting theory contains automatically gravity ⊕ since sl(2,R) sp(4). ⊂ The possibility to have a finite number of higher spin gauge fields is in contrast with D 3 where one needs an infinite number of > higher spin gauge fields to get a consistent theory. But what is the spin content ? Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 2 field. The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).
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 More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by ⊕ osp(1 4) osp(1 4), the bosonic subalgebra of which is | ⊕ | sp(4) sp(4). The resulting theory contains automatically gravity ⊕ since sl(2,R) sp(4). ⊂ The possibility to have a finite number of higher spin gauge fields is in contrast with D 3 where one needs an infinite number of > higher spin gauge fields to get a consistent theory. But what is the spin content ? Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 2 field. The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
10 / 28 The possibility to have a finite number of higher spin gauge fields is in contrast with D 3 where one needs an infinite number of > higher spin gauge fields to get a consistent theory. But what is the spin content ? Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 2 field. The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). Introduction
Three- More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by dimensional pure ⊕ gravity with Λ 0 osp(1 4) osp(1 4), the bosonic subalgebra of which is < | ⊕ | Hypergravity sp(4) sp(4). The resulting theory contains automatically gravity ⊕ Charges and since sl(2,R) sp(4). asymptotic ⊂ analysis
Hypersymmetry bounds
Black holes
Conclusions
10 / 28 Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 2 field. The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). Introduction
Three- More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by dimensional pure ⊕ gravity with Λ 0 osp(1 4) osp(1 4), the bosonic subalgebra of which is < | ⊕ | Hypergravity sp(4) sp(4). The resulting theory contains automatically gravity ⊕ Charges and since sl(2,R) sp(4). asymptotic ⊂ analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry bounds is in contrast with D 3 where one needs an infinite number of > Black holes higher spin gauge fields to get a consistent theory. But what is the Conclusions spin content ?
10 / 28 The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). Introduction
Three- More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by dimensional pure ⊕ gravity with Λ 0 osp(1 4) osp(1 4), the bosonic subalgebra of which is < | ⊕ | Hypergravity sp(4) sp(4). The resulting theory contains automatically gravity ⊕ Charges and since sl(2,R) sp(4). asymptotic ⊂ analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry bounds is in contrast with D 3 where one needs an infinite number of > Black holes higher spin gauge fields to get a consistent theory. But what is the Conclusions spin content ? Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 2 field.
10 / 28 Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The case of interest to us is obtained by replacing sl(2,R) by osp(1,4). Introduction
Three- More precisely, one replaces the gauge algebra sl(2,R) sl(2,R) by dimensional pure ⊕ gravity with Λ 0 osp(1 4) osp(1 4), the bosonic subalgebra of which is < | ⊕ | Hypergravity sp(4) sp(4). The resulting theory contains automatically gravity ⊕ Charges and since sl(2,R) sp(4). asymptotic ⊂ analysis The possibility to have a finite number of higher spin gauge fields Hypersymmetry bounds is in contrast with D 3 where one needs an infinite number of > Black holes higher spin gauge fields to get a consistent theory. But what is the Conclusions spin content ? Assuming principal embedding of sl(2,R) in sp(4), one gets one 5 spin-2 field, one spin-4 field and one spin- 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 black holes in 2+1 Basis of osp(1 4) : gravity | Marc Henneaux h i ¡ ¢ Li,Lj i j Li j , = − + Introduction £ ¤ L ,Um (3i m)U , i = − i m Three- µ ¶ + dimensional pure £ ¤ 3 gravity with Λ 0 Li,Sp i p Si p , < = 2 − + Hypergravity µ µ ¶ ¶ 1 ³ 2 2 ´ 2 2 2 2 Charges and [U ,U ] (m n) m n 4 m n mn 9 (mn 6)mn L m n 2 m n asymptotic = 2 3 − + − + − 3 − − 3 − + analysis 1 ³ 2 2 ´ (m n) m mn n 7 Um n , Hypersymmetry + 6 − − + − + bounds £ ¤ 1 ³ 3 2 2 3´ Black holes Um,Sp 2m 8m p 20mp 82p 23m 40p Si p , = 233 − + + − − + Conclusions © ª 1 ³ 2 2 ´ Sp,Sq Up q 6p 8pq 6q 9 Lp q . = + + 223 − + − + Here L , with i 0, 1, stand for the spin-2 generators that span the i = ± gravitational sl (2,R) subalgebra, while Um and Sp, with 1 3 m 0, 1, 2, 3 and p 2 , 2 , correspond to the spin-4 and = ± ± ± 5 = ± ± fermionic spin- 2 generators, respectively.
11 / 28 The action is £ ¤ I I A+ I [A−] = CS − CS
with Z · ¸ k4 2 3 ICS [A] str AdA A . = 4π + 3
Here, the level, k k/10, is expressed in terms of the Newton 4 = constant and the AdS radius according to k `/4G. = str [ ] stands for the supertrace of the fundamental (5 5) ··· × matrix representation. The connection reads
i m p A+ A L B U ψ S = µ i + µ m + µ p
and a similar expression holds for A−.
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 with Z · ¸ k4 2 3 ICS [A] str AdA A . = 4π + 3
Here, the level, k k/10, is expressed in terms of the Newton 4 = constant and the AdS radius according to k `/4G. = str [ ] stands for the supertrace of the fundamental (5 5) ··· × matrix representation. The connection reads
i m p A+ A L B U ψ S = µ i + µ m + µ p
and a similar expression holds for A−.
Dynamics
Hypersymmetric black holes in 2+1 The action is gravity £ ¤ Marc Henneaux I I A+ I [A−] = CS − CS Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
12 / 28 str [ ] stands for the supertrace of the fundamental (5 5) ··· × matrix representation. The connection reads
i m p A+ A L B U ψ S = µ i + µ m + µ p
and a similar expression holds for A−.
Dynamics
Hypersymmetric black holes in 2+1 The action is gravity £ ¤ Marc Henneaux I I A+ I [A−] = CS − CS Introduction
Three- dimensional pure with gravity with Λ 0 Z · ¸ < k4 2 3 Hypergravity ICS [A] str AdA A . = 4π + 3 Charges and asymptotic analysis Hypersymmetry Here, the level, k4 k/10, is expressed in terms of the Newton bounds = constant and the AdS radius according to k `/4G. Black holes = Conclusions
12 / 28 The connection reads
i m p A+ A L B U ψ S = µ i + µ m + µ p
and a similar expression holds for A−.
Dynamics
Hypersymmetric black holes in 2+1 The action is gravity £ ¤ Marc Henneaux I I A+ I [A−] = CS − CS Introduction
Three- dimensional pure with gravity with Λ 0 Z · ¸ < k4 2 3 Hypergravity ICS [A] str AdA A . = 4π + 3 Charges and asymptotic analysis Hypersymmetry Here, the level, k4 k/10, is expressed in terms of the Newton bounds = constant and the AdS radius according to k `/4G. Black holes = Conclusions str [ ] stands for the supertrace of the fundamental (5 5) ··· × matrix representation.
12 / 28 Dynamics
Hypersymmetric black holes in 2+1 The action is gravity £ ¤ Marc Henneaux I I A+ I [A−] = CS − CS Introduction
Three- dimensional pure with gravity with Λ 0 Z · ¸ < k4 2 3 Hypergravity ICS [A] str AdA A . = 4π + 3 Charges and asymptotic analysis Hypersymmetry Here, the level, k4 k/10, is expressed in terms of the Newton bounds = constant and the AdS radius according to k `/4G. Black holes = Conclusions str [ ] stands for the supertrace of the fundamental (5 5) ··· × matrix representation. The connection reads
i m p A+ A L B U ψ S = µ i + µ m + µ p
and a similar expression holds for A−.
12 / 28 In terms of the two osp(1 4) connections A+ and A−, the metric | and spin-4 field are defined by
g str(e e ), h str(e e e e ) astr(e e )str(e e ), µν ∼ µ ν µνρσ ∼ µ ν ρ σ + (µ ν ρ σ)
where e A+ A−, µ ∼ µ − µ p and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). 2 µa µa = µa ∼ µ E F 5 I The action is S[g ,h ,ψ ] S S S 2 S µν µνρσ µa = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
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 g str(e e ), h str(e e e e ) astr(e e )str(e e ), µν ∼ µ ν µνρσ ∼ µ ν ρ σ + (µ ν ρ σ)
where e A+ A−, µ ∼ µ − µ p and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). 2 µa µa = µa ∼ µ E F 5 I The action is S[g ,h ,ψ ] S S S 2 S µν µνρσ µa = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
13 / 28 where e A+ A−, µ ∼ µ − µ p and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). 2 µa µa = µa ∼ µ E F 5 I The action is S[g ,h ,ψ ] S S S 2 S µν µνρσ µa = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
13 / 28 p and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). 2 µa µa = µa ∼ µ E F 5 I The action is S[g ,h ,ψ ] S S S 2 S µν µνρσ µa = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis where e A+ A−, µ ∼ µ − µ Hypersymmetry bounds
Black holes
Conclusions
13 / 28 E F 5 I The action is S[g ,h ,ψ ] S S S 2 S µν µνρσ µa = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis where e A+ A−, µ ∼ µ − µ p Hypersymmetry and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). bounds 2 µa µa = µa ∼ µ Black holes
Conclusions
13 / 28 where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis where e A+ A−, µ ∼ µ − µ p Hypersymmetry and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). bounds 2 µa µa = µa ∼ µ Black holes E F 5 I The action is S[gµν,hµνρσ,ψµa] S S S 2 S Conclusions = + + +
13 / 28 These interaction terms are not known in closed form. They can be constructed perturbatively.
Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis where e A+ A−, µ ∼ µ − µ p Hypersymmetry and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). bounds 2 µa µa = µa ∼ µ Black holes E F 5 I The action is S[gµν,hµνρσ,ψµa] S S S 2 S Conclusions = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI stands for the higher order interaction terms necessary to make the theory consistent.
13 / 28 Dynamics
Hypersymmetric black holes in 2+1 In terms of the two osp(1 4) connections A+ and A−, the metric gravity | Marc Henneaux and spin-4 field are defined by
Introduction
Three- dimensional pure gravity with Λ 0 gµν str(eµeν), hµνρσ str(eµeνeρeσ) astr(e(µeν)str(eρeσ)), < ∼ ∼ + Hypergravity
Charges and asymptotic analysis where e A+ A−, µ ∼ µ − µ p Hypersymmetry and the spin- 5 field is ψ , γaψ 0 (ψα ψ ). bounds 2 µa µa = µa ∼ µ Black holes E F 5 I The action is S[gµν,hµνρσ,ψµa] S S S 2 S Conclusions = + + + where SE is the Einstein action, SF the (covariantized) Fronsdal 5 2 5 action for a spin-4 field, S the (covariantized) spin- 2 action and SI 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 An important (and puzzling !) feature of higher spin gauge theories is that the metric gµν transforms under the gauge transformations of the spin-4 gauge field hλµνρ. There is no known definition of a geometry that would be invariant under higher spin gauge symmetries. In particular, given a solution to the field equation, there is no known way to ascribe to it a well-defined causal structure. We shall come back to that question later.
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 There is no known definition of a geometry that would be invariant under higher spin gauge symmetries. In particular, given a solution to the field equation, there is no known way to ascribe to it a well-defined causal structure. We shall come back to that question later.
Absence of a well-defined geometry
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure An important (and puzzling !) feature of higher spin gauge gravity with Λ 0 < theories is that the metric gµν transforms under the gauge Hypergravity transformations of the spin-4 gauge field hλµνρ. Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
14 / 28 In particular, given a solution to the field equation, there is no known way to ascribe to it a well-defined causal structure. We shall come back to that question later.
Absence of a well-defined geometry
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure An important (and puzzling !) feature of higher spin gauge gravity with Λ 0 < theories is that the metric gµν transforms under the gauge Hypergravity transformations of the spin-4 gauge field hλµνρ. Charges and asymptotic There is no known definition of a geometry that would be analysis
Hypersymmetry invariant under higher spin gauge symmetries. bounds
Black holes
Conclusions
14 / 28 We shall come back to that question later.
Absence of a well-defined geometry
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure An important (and puzzling !) feature of higher spin gauge gravity with Λ 0 < theories is that the metric gµν transforms under the gauge Hypergravity transformations of the spin-4 gauge field hλµνρ. Charges and asymptotic There is no known definition of a geometry that would be analysis
Hypersymmetry invariant under higher spin gauge symmetries. 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
Three- dimensional pure An important (and puzzling !) feature of higher spin gauge gravity with Λ 0 < theories is that the metric gµν transforms under the gauge Hypergravity transformations of the spin-4 gauge field hλµνρ. Charges and asymptotic There is no known definition of a geometry that would be analysis
Hypersymmetry invariant under higher spin gauge symmetries. 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 The boundary conditions were first investigated in the metric formulation and a precise definition of what is meant by “asymptotically anti-de Sitter metric" was given. These boundary conditions can be reformulated in terms of the Chern-Simons connection. It turns out that (in a suitable gauge) they take exactly the same form as the so-called Drinfeld-Sokolov Hamiltonian reduction conditions, namely
µ ¶ ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
Coussaert, Henneaux, van Driel 1995
Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
15 / 28 These boundary conditions can be reformulated in terms of the Chern-Simons connection. It turns out that (in a suitable gauge) they take exactly the same form as the so-called Drinfeld-Sokolov Hamiltonian reduction conditions, namely
µ ¶ ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
Coussaert, Henneaux, van Driel 1995
Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux The boundary conditions were first investigated in the metric 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 It turns out that (in a suitable gauge) they take exactly the same form as the so-called Drinfeld-Sokolov Hamiltonian reduction conditions, namely
µ ¶ ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
Coussaert, Henneaux, van Driel 1995
Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux The boundary conditions were first investigated in the metric 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 These boundary conditions can be reformulated in terms of the < Hypergravity Chern-Simons connection.
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
15 / 28 µ ¶ ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
Coussaert, Henneaux, van Driel 1995
Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux The boundary conditions were first investigated in the metric 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 These boundary conditions can be reformulated in terms of the < Hypergravity Chern-Simons connection.
Charges and asymptotic It turns out that (in a suitable gauge) they take exactly the same analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry bounds conditions, namely
Black holes
Conclusions
15 / 28 Coussaert, Henneaux, van Driel 1995
Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux The boundary conditions were first investigated in the metric 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 These boundary conditions can be reformulated in terms of the < Hypergravity Chern-Simons connection.
Charges and asymptotic It turns out that (in a suitable gauge) they take exactly the same analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry bounds conditions, namely
Black holes µ ¶ Conclusions ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
15 / 28 Boundary conditions - Pure gravity
Hypersymmetric black holes in 2+1 We first consider pure gravity gravity
Marc Henneaux The boundary conditions were first investigated in the metric 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 These boundary conditions can be reformulated in terms of the < Hypergravity Chern-Simons connection.
Charges and asymptotic It turns out that (in a suitable gauge) they take exactly the same analysis form as the so-called Drinfeld-Sokolov Hamiltonian reduction Hypersymmetry bounds conditions, namely
Black holes µ ¶ Conclusions ¡ ¢ 2π ¡ ¢ 1 Aϕ± r,ϕ L 1 L ± ϕ L 1 O , r ± ∓ −→→∞ − k + r and µ 1 ¶ A± O . r r −→→∞ r
Coussaert, Henneaux, van Driel 1995
15 / 28 The “asymptotic symmetries" are those gauge transformations δA± ∂ Λ± [A±,Λ±] that preserve the boundary conditions, i.e., i = i + i such that A± δA± fulfills also the boundary conditions. i + i They are given by µ ¶ ¡ ¢ 2π ¡ ¢ Λ± ² ϕ L 1 L ± ϕ L 1 r ± ± ∓ −→→∞ ± − k ¡ ¢ 1 ¡ ¢ ²0 ϕ L0 ²00 ϕ L 1 ∓ ± ± 2 ± ∓
The functions ² ¡ϕ¢ are arbitrary functions of ϕ and parametrize ± the asymptotic symmetries.
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 They are given by µ ¶ ¡ ¢ 2π ¡ ¢ Λ± ² ϕ L 1 L ± ϕ L 1 r ± ± ∓ −→→∞ ± − k ¡ ¢ 1 ¡ ¢ ²0 ϕ L0 ²00 ϕ L 1 ∓ ± ± 2 ± ∓
The functions ² ¡ϕ¢ are arbitrary functions of ϕ and parametrize ± the asymptotic symmetries.
Asymptotic symmetries
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The “asymptotic symmetries" are those gauge transformations
Three- δAi± ∂iΛ± [Ai±,Λ±] that preserve the boundary conditions, i.e., dimensional pure = + gravity with Λ 0 such that A± δA± fulfills also the boundary conditions. < i + i Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
16 / 28 The functions ² ¡ϕ¢ are arbitrary functions of ϕ and parametrize ± the asymptotic symmetries.
Asymptotic symmetries
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The “asymptotic symmetries" are those gauge transformations
Three- δAi± ∂iΛ± [Ai±,Λ±] that preserve the boundary conditions, i.e., dimensional pure = + gravity with Λ 0 such that A± δA± fulfills also the boundary conditions. < i + i Hypergravity They are given by Charges and asymptotic µ ¶ analysis ¡ ¢ 2π ¡ ¢ Λ± ² ϕ L 1 L ± ϕ L 1 Hypersymmetry r ± ± ∓ bounds −→→∞ ± − k Black holes ¡ ¢ 1 ¡ ¢ ²0 ϕ L0 ²00 ϕ L 1 Conclusions ∓ ± ± 2 ± ∓
16 / 28 Asymptotic symmetries
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction The “asymptotic symmetries" are those gauge transformations
Three- δAi± ∂iΛ± [Ai±,Λ±] that preserve the boundary conditions, i.e., dimensional pure = + gravity with Λ 0 such that A± δA± fulfills also the boundary conditions. < i + i Hypergravity They are given by Charges and asymptotic µ ¶ analysis ¡ ¢ 2π ¡ ¢ Λ± ² ϕ L 1 L ± ϕ L 1 Hypersymmetry r ± ± ∓ bounds −→→∞ ± − k Black holes ¡ ¢ 1 ¡ ¢ ²0 ϕ L0 ²00 ϕ L 1 Conclusions ∓ ± ± 2 ± ∓
The functions ² ¡ϕ¢ are arbitrary functions of ϕ and parametrize ± the asymptotic symmetries.
16 / 28 Furthermore, one easily finds that the generators of the asymptotic symmetry algebra are given explictly by the L ±’s themselves (when the constraints hold) and read explicitly Z ¡ ¢ ¡ ¢ Q [² ] ² ϕ L ± ϕ dϕ ± ± = ± r ± →∞ These generators obey the Virasoro algebra. More precisely, the Fourier components Ln± obey, in terms of the Poisson bracket, the Virasoro algebra with the classical central charge c 6k 3`/2G, = =
£ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
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 These generators obey the Virasoro algebra. More precisely, the Fourier components Ln± obey, in terms of the Poisson bracket, the Virasoro algebra with the classical central charge c 6k 3`/2G, = =
£ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
17 / 28 More precisely, the Fourier components Ln± obey, in terms of the Poisson bracket, the Virasoro algebra with the classical central charge c 6k 3`/2G, = =
£ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and These generators obey the Virasoro algebra. asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
17 / 28 £ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and These generators obey the Virasoro algebra. asymptotic analysis More precisely, the Fourier components Ln± obey, in terms of the Hypersymmetry Poisson bracket, the Virasoro algebra with the classical central bounds charge c 6k 3`/2G, Black holes = = Conclusions
17 / 28 £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and These generators obey the Virasoro algebra. asymptotic analysis More precisely, the Fourier components Ln± obey, in terms of the Hypersymmetry Poisson bracket, the Virasoro algebra with the classical central bounds charge c 6k 3`/2G, Black holes = = Conclusions £ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 +
17 / 28 Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and These generators obey the Virasoro algebra. asymptotic analysis More precisely, the Fourier components Ln± obey, in terms of the Hypersymmetry Poisson bracket, the Virasoro algebra with the classical central bounds charge c 6k 3`/2G, Black holes = = Conclusions £ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra).
17 / 28 Generators of asymptotic symmetries - Virasoro algebra
Hypersymmetric black holes in 2+1 Furthermore, one easily finds that the generators of the gravity asymptotic symmetry algebra are given explictly by the L ’s Marc Henneaux ± themselves (when the constraints hold) and read explicitly Introduction Z Three- ¡ ¢ ¡ ¢ dimensional pure Q [² ] ² ϕ L ± ϕ dϕ gravity with Λ 0 ± ± = ± r ± < →∞ Hypergravity
Charges and These generators obey the Virasoro algebra. asymptotic analysis More precisely, the Fourier components Ln± obey, in terms of the Hypersymmetry Poisson bracket, the Virasoro algebra with the classical central bounds charge c 6k 3`/2G, Black holes = = Conclusions £ ¤ k 3 i Lm±,Ln± PB (m n)Lm± n m δm n,0. = − + + 2 + £ ¤ and commute between themselves L +,L − 0 (2D m n PB = conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.
17 / 28 The same asymptotic analysis can be performed for hypergravity. One gets an enhancement of the asymptotic algebra, from the Virasoro algebra to the W 5 -superalgebra, which contains the (2, 2 ,4) Virasoro generators but also generators of higher conformal weights. How does this proceed ? Sugra : Henneaux, Maoz, Schwimmer (2000) ; Higher spins : S.-J. Rey + MH (2010) ; A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen (2010)
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 One gets an enhancement of the asymptotic algebra, from the Virasoro algebra to the W 5 -superalgebra, which contains the (2, 2 ,4) Virasoro generators but also generators of higher conformal weights. How does this proceed ? Sugra : Henneaux, Maoz, Schwimmer (2000) ; Higher spins : S.-J. Rey + MH (2010) ; A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen (2010)
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- The same asymptotic analysis can be performed for hypergravity. dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
18 / 28 How does this proceed ? Sugra : Henneaux, Maoz, Schwimmer (2000) ; Higher spins : S.-J. Rey + MH (2010) ; A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen (2010)
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- The same asymptotic analysis can be performed for hypergravity. dimensional pure gravity with Λ 0 One gets an enhancement of the asymptotic algebra, from the < Hypergravity Virasoro algebra to the W 5 -superalgebra, which contains the (2, 2 ,4) Charges and asymptotic Virasoro generators but also generators of higher conformal analysis weights. Hypersymmetry bounds
Black holes
Conclusions
18 / 28 Sugra : Henneaux, Maoz, Schwimmer (2000) ; Higher spins : S.-J. Rey + MH (2010) ; A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen (2010)
Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- The same asymptotic analysis can be performed for hypergravity. dimensional pure gravity with Λ 0 One gets an enhancement of the asymptotic algebra, from the < Hypergravity Virasoro algebra to the W 5 -superalgebra, which contains the (2, 2 ,4) Charges and asymptotic Virasoro generators but also generators of higher conformal analysis weights. Hypersymmetry bounds How does this proceed ? Black holes
Conclusions
18 / 28 Hypergravity
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- The same asymptotic analysis can be performed for hypergravity. dimensional pure gravity with Λ 0 One gets an enhancement of the asymptotic algebra, from the < Hypergravity Virasoro algebra to the W 5 -superalgebra, which contains the (2, 2 ,4) Charges and asymptotic Virasoro generators but also generators of higher conformal analysis weights. Hypersymmetry bounds How does this proceed ? 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 The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type
¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , r ± ∓ ∓ 2 −→→∞ − k + 5k − k ∓
¡ ¢ ¡ ¢ Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear along the lowest (highest)-weight generators. These boundary conditions are invariant under gauge transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
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 ¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , r ± ∓ ∓ 2 −→→∞ − k + 5k − k ∓
¡ ¢ ¡ ¢ Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear along the lowest (highest)-weight generators. These boundary conditions are invariant under gauge transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
Boundary conditions
Hypersymmetric black holes in 2+1 The asymptotic conditions that generalize those found for pure gravity
Marc Henneaux gravity are again of Drinfeld-Sokolov type
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
19 / 28 ¡ ¢ ¡ ¢ Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear along the lowest (highest)-weight generators. These boundary conditions are invariant under gauge transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
Boundary conditions
Hypersymmetric black holes in 2+1 The asymptotic conditions that generalize those found for pure gravity
Marc Henneaux gravity are again of Drinfeld-Sokolov type
Introduction
Three- dimensional pure ¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ gravity with Λ 0 Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , < r−→ ± − k ∓ + 5k ∓ − k ∓ 2 Hypergravity →∞
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
19 / 28 These boundary conditions are invariant under gauge transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
Boundary conditions
Hypersymmetric black holes in 2+1 The asymptotic conditions that generalize those found for pure gravity
Marc Henneaux gravity are again of Drinfeld-Sokolov type
Introduction
Three- dimensional pure ¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ gravity with Λ 0 Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , < r−→ ± − k ∓ + 5k ∓ − k ∓ 2 Hypergravity →∞
Charges and asymptotic ¡ ¢ ¡ ¢ analysis Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear
Hypersymmetry along the lowest (highest)-weight generators. bounds
Black holes
Conclusions
19 / 28 with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
Boundary conditions
Hypersymmetric black holes in 2+1 The asymptotic conditions that generalize those found for pure gravity
Marc Henneaux gravity are again of Drinfeld-Sokolov type
Introduction
Three- dimensional pure ¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ gravity with Λ 0 Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , < r−→ ± − k ∓ + 5k ∓ − k ∓ 2 Hypergravity →∞
Charges and asymptotic ¡ ¢ ¡ ¢ analysis Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear
Hypersymmetry along the lowest (highest)-weight generators. bounds
Black holes These boundary conditions are invariant under gauge
Conclusions transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
19 / 28 Boundary conditions
Hypersymmetric black holes in 2+1 The asymptotic conditions that generalize those found for pure gravity
Marc Henneaux gravity are again of Drinfeld-Sokolov type
Introduction
Three- dimensional pure ¡ ¢ 2π ¡ ¢ π ¡ ¢ 2π ¡ ¢ gravity with Λ 0 Aϕ± r,ϕ L 1 L ± ϕ L 1 U ± ϕ U 3 ψ± ϕ S 3 , < r−→ ± − k ∓ + 5k ∓ − k ∓ 2 Hypergravity →∞
Charges and asymptotic ¡ ¢ ¡ ¢ analysis Again, the non trivial fields L ± ϕ , U ± ϕ and ψ±(ϕ) appear
Hypersymmetry along the lowest (highest)-weight generators. bounds
Black holes These boundary conditions are invariant under gauge
Conclusions transformations that are generated by Z £ ¤ ¡ ¢ Q± ² ,χ ,ϑ dϕ ² L ± χ U ± iϑ ψ± , ± ± ± = ± ± + ± − ±
with ² , χ and ϑ arbitrary functions of ϕ. The “charges" ¡ ±¢ ± ¡ ¢ ± L ± ϕ , U ± ϕ and ψ±(ϕ) form the W 5 -superalgebra. (2, 2 ,4)
19 / 28 W 5 -superalgebra (2, 2 ,4)
Hypersymmetric black holes in 2+1 gravity k 3 0 i[Lm,Ln]PB (m n)Lm n m δm n , Marc Henneaux = − + + 2 + i[Lm,Un]PB (3m n)Um n , Introduction = − + µ ¶ Three- £ ¤ 3 i Lm,ψn m n ψm n , dimensional pure PB = 2 − + gravity with Λ 0 < 1 ³ 4 3 2 2 3 4´ Hypergravity i[Um,Un]PB (m n) 3m 2m n 4m n 2mn 3n Lm n = 2232 − − + − + + Charges and asymptotic 3 1 ³ 2 2´ 2 3π (6) analysis (m n) m mn n Um n (m n)Λm n + 6 − − + + − k − + Hypersymmetry 2 bounds 7 π ³ ´ k (m n) m2 4mn n2 Λ(4) m7δ0 , 2 m n 3 2 m n Black holes − 3 k − + + + + 2 3 + Conclusions 1 ³ ´ 23π i£U ,ψ ¤ m3 4m2n 10mn2 20n3 ψ iΛ(11/2) m n PB 2 m n m n = 2 3 − + − + − 3k + π (9/2) (23m 82n)Λm n , + 3k − + µ ¶ £ ¤ 1 2 4 2 3π (4) k 4 0 i ψm,ψn PB Um n m mn n Lm n Λm n m δm n , = + + 2 − 3 + + + k + + 6 + 5 The generators Un, ψn have respective conformal weights 4 and 2 ; 3` unchanged central charge c 6k 2G (two copies). = = 20 / 28 The fermions can be anti-periodic or periodic. We assume here that they are periodic as this is the case relevant to black holes. We also assume that they are only zero-modes (“rest frame"). £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 PB reads
1 ¡ ¢ 2 2 3π 2 (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ U L 0. + = 2π 0 = + k ≥
(The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
Hypersymmetry bounds
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
21 / 28 We assume here that they are periodic as this is the case relevant to black holes. We also assume that they are only zero-modes (“rest frame"). £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 PB reads
1 ¡ ¢ 2 2 3π 2 (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ U L 0. + = 2π 0 = + k ≥
(The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
21 / 28 £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 PB reads
1 ¡ ¢ 2 2 3π 2 (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ U L 0. + = 2π 0 = + k ≥
(The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction We assume here that they are periodic as this is the case relevant Three- to black holes. We also assume that they are only zero-modes dimensional pure gravity with Λ 0 < (“rest frame"). Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
21 / 28 1 ¡ ¢ 2 2 3π 2 (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ U L 0. + = 2π 0 = + k ≥
(The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction We assume here that they are periodic as this is the case relevant Three- to black holes. We also assume that they are only zero-modes dimensional pure gravity with Λ 0 < (“rest frame"). Hypergravity £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 reads Charges and PB asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
21 / 28 (The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction We assume here that they are periodic as this is the case relevant Three- to black holes. We also assume that they are only zero-modes dimensional pure gravity with Λ 0 < (“rest frame"). Hypergravity £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 reads Charges and PB asymptotic analysis 1 ¡ ¢ 2 2 3π 2 Hypersymmetry (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ 0 U L 0. bounds + = 2π = + k ≥ Black holes
Conclusions
21 / 28 This is a nonlinear bound.
Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction We assume here that they are periodic as this is the case relevant Three- to black holes. We also assume that they are only zero-modes dimensional pure gravity with Λ 0 < (“rest frame"). Hypergravity £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 reads Charges and PB asymptotic analysis 1 ¡ ¢ 2 2 3π 2 Hypersymmetry (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ 0 U L 0. bounds + = 2π = + k ≥ Black holes Conclusions (The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.)
21 / 28 Hypersymmetry bounds
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux The fermions can be anti-periodic or periodic.
Introduction We assume here that they are periodic as this is the case relevant Three- to black holes. We also assume that they are only zero-modes dimensional pure gravity with Λ 0 < (“rest frame"). Hypergravity £ ¤ The quantum version of the Poisson bracket ψ0,ψ0 reads Charges and PB asymptotic analysis 1 ¡ ¢ 2 2 3π 2 Hypersymmetry (2π)− ψˆ 0ψˆ 0 ψˆ 0ψˆ 0 ψˆ 0 U L 0. bounds + = 2π = + k ≥ Black holes Conclusions (The quantum W (2,5/2,4)-superalgebra, with the unitarity † † † conditions Lm L m, Um U m, ψm ψ m implied by the = − = − = − classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.
21 / 28 In the absence of a well-defined geometry, black holes and black hole thermodynamics are defined through the Euclidean continuation. Gutperle, Kraus (2011) ; Ammon, Gutperle, Kraus, Perlmutter (2011, 2013) The Euclidean BTZ black hole has solid torus topology. Carlip, Teitelboim (1995)
Black holes - Euclidean continuation
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
22 / 28 Gutperle, Kraus (2011) ; Ammon, Gutperle, Kraus, Perlmutter (2011, 2013) The Euclidean BTZ black hole has solid torus topology. Carlip, Teitelboim (1995)
Black holes - Euclidean continuation
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure In the absence of a well-defined geometry, black holes and black gravity with Λ 0 < Hypergravity hole thermodynamics are defined through the Euclidean
Charges and continuation. asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
22 / 28 The Euclidean BTZ black hole has solid torus topology. Carlip, Teitelboim (1995)
Black holes - Euclidean continuation
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure In the absence of a well-defined geometry, black holes and black gravity with Λ 0 < Hypergravity hole thermodynamics are defined through the Euclidean
Charges and continuation. asymptotic analysis Gutperle, Kraus (2011) ; Ammon, Gutperle, Kraus, Perlmutter Hypersymmetry (2011, 2013) bounds
Black holes
Conclusions
22 / 28 Carlip, Teitelboim (1995)
Black holes - Euclidean continuation
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure In the absence of a well-defined geometry, black holes and black gravity with Λ 0 < Hypergravity hole thermodynamics are defined through the Euclidean
Charges and continuation. asymptotic analysis Gutperle, Kraus (2011) ; Ammon, Gutperle, Kraus, Perlmutter Hypersymmetry (2011, 2013) bounds
Black holes The Euclidean BTZ black hole has solid torus topology.
Conclusions
22 / 28 Black holes - Euclidean continuation
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure In the absence of a well-defined geometry, black holes and black gravity with Λ 0 < Hypergravity hole thermodynamics are defined through the Euclidean
Charges and continuation. asymptotic analysis Gutperle, Kraus (2011) ; Ammon, Gutperle, Kraus, Perlmutter Hypersymmetry (2011, 2013) bounds
Black holes The Euclidean BTZ black hole has solid torus topology. Conclusions Carlip, Teitelboim (1995)
22 / 28 The topology of the Euclidean black hole is a solid torus, R2 S1. The “Euclidean horizon” r × + is the origin of a system of polar coordinates r,τ in R2. The Euclidean time τ is the polar angle. On the other hand, the S1 is parametrized by the angle ϕ.
Black hole topology - Euclidean formulation
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
23 / 28 The topology of the Euclidean black hole is a solid torus, R2 S1. The “Euclidean horizon” r × + is the origin of a system of polar coordinates r,τ in R2. The Euclidean time τ is the polar angle. On the other hand, the S1 is parametrized by the angle ϕ.
Black hole topology - Euclidean formulation
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
23 / 28 Black hole topology - Euclidean formulation
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
The topology of the Euclidean black hole is a solid torus, R2 S1. The “Euclidean horizon” r × + is the origin of a system of polar coordinates r,τ in R2. The Euclidean time τ is the polar angle. On the other hand, the S1 is parametrized by the angle ϕ.
23 / 28 The (Euclidean) black hole is the most general flat connection (with Euclidean version of the algebra) on a solid torus with no singularity, obeying the appropriate boundary conditions at infinity, and allowing for a consistent thermodynamics (real entropy). One can derive the whole thermodynamics and in particular the below extremality condition (existence of a horizon) within the Chern-Simons formulation, without invoking the explicit form of the metric or even metric concepts (causal structure etc). This approach is crucial when higher spins are included, where there is no well-defined geometry.
Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
24 / 28 and allowing for a consistent thermodynamics (real entropy). One can derive the whole thermodynamics and in particular the below extremality condition (existence of a horizon) within the Chern-Simons formulation, without invoking the explicit form of the metric or even metric concepts (causal structure etc). This approach is crucial when higher spins are included, where there is no well-defined geometry.
Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction The (Euclidean) black hole is the most general flat connection Three- (with Euclidean version of the algebra) on a solid torus with no dimensional pure gravity with Λ 0 < singularity, obeying the appropriate boundary conditions at Hypergravity infinity, Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
24 / 28 One can derive the whole thermodynamics and in particular the below extremality condition (existence of a horizon) within the Chern-Simons formulation, without invoking the explicit form of the metric or even metric concepts (causal structure etc). This approach is crucial when higher spins are included, where there is no well-defined geometry.
Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction The (Euclidean) black hole is the most general flat connection Three- (with Euclidean version of the algebra) on a solid torus with no dimensional pure gravity with Λ 0 < singularity, obeying the appropriate boundary conditions at Hypergravity infinity, Charges and asymptotic and allowing for a consistent thermodynamics (real entropy). analysis
Hypersymmetry bounds
Black holes
Conclusions
24 / 28 without invoking the explicit form of the metric or even metric concepts (causal structure etc). This approach is crucial when higher spins are included, where there is no well-defined geometry.
Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction The (Euclidean) black hole is the most general flat connection Three- (with Euclidean version of the algebra) on a solid torus with no dimensional pure gravity with Λ 0 < singularity, obeying the appropriate boundary conditions at Hypergravity infinity, Charges and asymptotic and allowing for a consistent thermodynamics (real entropy). analysis
Hypersymmetry One can derive the whole thermodynamics and in particular the bounds below extremality condition (existence of a horizon) within the Black holes Chern-Simons formulation, Conclusions
24 / 28 This approach is crucial when higher spins are included, where there is no well-defined geometry.
Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction The (Euclidean) black hole is the most general flat connection Three- (with Euclidean version of the algebra) on a solid torus with no dimensional pure gravity with Λ 0 < singularity, obeying the appropriate boundary conditions at Hypergravity infinity, Charges and asymptotic and allowing for a consistent thermodynamics (real entropy). analysis
Hypersymmetry One can derive the whole thermodynamics and in particular the bounds below extremality condition (existence of a horizon) within the Black holes Chern-Simons formulation, Conclusions without invoking the explicit form of the metric or even metric concepts (causal structure etc).
24 / 28 Black hole in Chern-Simons formulation - Definition
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux More precisely :
Introduction The (Euclidean) black hole is the most general flat connection Three- (with Euclidean version of the algebra) on a solid torus with no dimensional pure gravity with Λ 0 < singularity, obeying the appropriate boundary conditions at Hypergravity infinity, Charges and asymptotic and allowing for a consistent thermodynamics (real entropy). analysis
Hypersymmetry One can derive the whole thermodynamics and in particular the bounds below extremality condition (existence of a horizon) within the Black holes Chern-Simons formulation, Conclusions without invoking the explicit form of the metric or even metric concepts (causal structure etc). This approach is crucial when higher spins are included, where there is no well-defined geometry.
24 / 28 The hypergravity Euclidean black hole is then a flat osp(1 4;C)-connection defined on the solid torus, | obeying the above boundary conditions and regular everywhere, including at the origin r . + This is the generalization of absence of conical singularity at the horizon. Such solutions exist.
Hypergravity black holes
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
25 / 28 obeying the above boundary conditions and regular everywhere, including at the origin r . + This is the generalization of absence of conical singularity at the horizon. Such solutions exist.
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The hypergravity Euclidean black hole is then a flat gravity with Λ 0 < osp(1 4;C)-connection defined on the solid torus, Hypergravity | Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
25 / 28 and regular everywhere, including at the origin r . + This is the generalization of absence of conical singularity at the horizon. Such solutions exist.
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The hypergravity Euclidean black hole is then a flat gravity with Λ 0 < osp(1 4;C)-connection defined on the solid torus, Hypergravity | Charges and obeying the above boundary conditions asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
25 / 28 This is the generalization of absence of conical singularity at the horizon. Such solutions exist.
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The hypergravity Euclidean black hole is then a flat gravity with Λ 0 < osp(1 4;C)-connection defined on the solid torus, Hypergravity | Charges and obeying the above boundary conditions asymptotic analysis and regular everywhere, including at the origin r . Hypersymmetry + bounds
Black holes
Conclusions
25 / 28 Such solutions exist.
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The hypergravity Euclidean black hole is then a flat gravity with Λ 0 < osp(1 4;C)-connection defined on the solid torus, Hypergravity | Charges and obeying the above boundary conditions asymptotic analysis and regular everywhere, including at the origin r . Hypersymmetry + bounds This is the generalization of absence of conical singularity at the Black holes horizon. Conclusions
25 / 28 Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity
Marc Henneaux
Introduction
Three- dimensional pure The hypergravity Euclidean black hole is then a flat gravity with Λ 0 < osp(1 4;C)-connection defined on the solid torus, Hypergravity | Charges and obeying the above boundary conditions asymptotic analysis and regular everywhere, including at the origin r . Hypersymmetry + bounds This is the generalization of absence of conical singularity at the Black holes horizon. Conclusions Such solutions exist.
25 / 28 The Euclidean connection for the black hole is explicitly given by
2π π Aϕ L1 L L 1 U U 3 , = − k − + 5k −
where L and U are now complex constants (related to the Lorentzian L ± and U ±) (mass, angular momentum and spin-4 charges).
The component Aτ along Euclidean time can be determined from the equations of motion and the boundary conditions, and involve two complex functions, ξ and µ (“chemical potentials"). The regularity condition (absence of conical singularity at the origin) determines ξ and µ in terms of the charges L , U .
Hypergravity black holes
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
26 / 28 2π π Aϕ L1 L L 1 U U 3 , = − k − + 5k −
where L and U are now complex constants (related to the Lorentzian L ± and U ±) (mass, angular momentum and spin-4 charges).
The component Aτ along Euclidean time can be determined from the equations of motion and the boundary conditions, and involve two complex functions, ξ and µ (“chemical potentials"). The regularity condition (absence of conical singularity at the origin) determines ξ and µ in terms of the charges L , U .
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity Marc Henneaux The Euclidean connection for the black hole is explicitly given by Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
26 / 28 where L and U are now complex constants (related to the Lorentzian L ± and U ±) (mass, angular momentum and spin-4 charges).
The component Aτ along Euclidean time can be determined from the equations of motion and the boundary conditions, and involve two complex functions, ξ and µ (“chemical potentials"). The regularity condition (absence of conical singularity at the origin) determines ξ and µ in terms of the charges L , U .
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity Marc Henneaux The Euclidean connection for the black hole is explicitly given by Introduction Three- 2π π dimensional pure A L L L U U , gravity with Λ 0 ϕ 1 1 3 < = − k − + 5k − Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
26 / 28 The component Aτ along Euclidean time can be determined from the equations of motion and the boundary conditions, and involve two complex functions, ξ and µ (“chemical potentials"). The regularity condition (absence of conical singularity at the origin) determines ξ and µ in terms of the charges L , U .
Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity Marc Henneaux The Euclidean connection for the black hole is explicitly given by Introduction Three- 2π π dimensional pure A L L L U U , gravity with Λ 0 ϕ 1 1 3 < = − k − + 5k − Hypergravity
Charges and asymptotic where L and U are now complex constants (related to the analysis Lorentzian and ) (mass, angular momentum and spin-4 Hypersymmetry L ± U ± bounds charges). Black holes
Conclusions
26 / 28 Hypergravity black holes
Hypersymmetric black holes in 2+1 gravity Marc Henneaux The Euclidean connection for the black hole is explicitly given by Introduction Three- 2π π dimensional pure A L L L U U , gravity with Λ 0 ϕ 1 1 3 < = − k − + 5k − Hypergravity
Charges and asymptotic where L and U are now complex constants (related to the analysis Lorentzian and ) (mass, angular momentum and spin-4 Hypersymmetry L ± U ± bounds charges). Black holes The component A along Euclidean time can be determined from Conclusions τ the equations of motion and the boundary conditions, and involve two complex functions, ξ and µ (“chemical potentials"). The regularity condition (absence of conical singularity at the origin) determines ξ and µ in terms of the charges L , U .
26 / 28 The Euclidean action gives the entropy and the thermodynamics can be consistently defined provided the charges are within the “extremal limit" corresponding to a real entropy. The expressions are rather cumbersome but the derivation is direct. k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
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
27 / 28 and the thermodynamics can be consistently defined provided the charges are within the “extremal limit" corresponding to a real entropy. The expressions are rather cumbersome but the derivation is direct. k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy
Marc Henneaux
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
27 / 28 provided the charges are within the “extremal limit" corresponding to a real entropy. The expressions are rather cumbersome but the derivation is direct. k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
27 / 28 corresponding to a real entropy. The expressions are rather cumbersome but the derivation is direct. k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
27 / 28 The expressions are rather cumbersome but the derivation is direct. k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
27 / 28 k 24 ¡ ¢2 One finds as extremality bounds L ± 0, U ± L ± and ≥ 3π ≤ 32
¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity The expressions are rather cumbersome but the derivation is Charges and direct. asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
27 / 28 ¡ ¢2 k L ± U ± − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity The expressions are rather cumbersome but the derivation is Charges and direct. asymptotic analysis k 24 ¡ ¢2 One finds as extremality bounds L ± 0, 3π U ± 2 L ± and Hypersymmetry ≥ ≤ 3 bounds
Black holes
Conclusions
27 / 28 This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity The expressions are rather cumbersome but the derivation is Charges and direct. asymptotic analysis k 24 ¡ ¢2 One finds as extremality bounds L ± 0, 3π U ± 2 L ± and Hypersymmetry ≥ ≤ 3 bounds Black holes ¡ ¢2 k L ± U ± Conclusions − ≤ 3π
27 / 28 The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity The expressions are rather cumbersome but the derivation is Charges and direct. asymptotic analysis k 24 ¡ ¢2 One finds as extremality bounds L ± 0, 3π U ± 2 L ± and Hypersymmetry ≥ ≤ 3 bounds Black holes ¡ ¢2 k L ± U ± Conclusions − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra.
27 / 28 Hypergravity black hole -Thermodynamics
Hypersymmetric black holes in 2+1 gravity The Euclidean action gives the entropy Marc Henneaux and the thermodynamics can be consistently defined Introduction provided the charges are within the “extremal limit" Three- dimensional pure gravity with Λ 0 corresponding to a real entropy. < Hypergravity The expressions are rather cumbersome but the derivation is Charges and direct. asymptotic analysis k 24 ¡ ¢2 One finds as extremality bounds L ± 0, 3π U ± 2 L ± and Hypersymmetry ≥ ≤ 3 bounds Black holes ¡ ¢2 k L ± U ± Conclusions − ≤ 3π
This last bound is just the hypersymmetric bound found above from the algebra. The black holes that saturate this bound are extremal and hypersymmetric (possess Killing vector-spinors).
27 / 28 Hypersymmetry bounds exist, are non trivial and are interesting. They provide nonlinear constraints on the bosonic charges. In the case of 2+1 hypergravity, the black holes that saturate the bounds are extremal. The whole discussion can be pursued purely algebraically, without invoking geometrical concepts. The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
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
28 / 28 They provide nonlinear constraints on the bosonic charges. In the case of 2+1 hypergravity, the black holes that saturate the bounds are extremal. The whole discussion can be pursued purely algebraically, without invoking geometrical concepts. The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting.
Introduction
Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
28 / 28 In the case of 2+1 hypergravity, the black holes that saturate the bounds are extremal. The whole discussion can be pursued purely algebraically, without invoking geometrical concepts. The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure gravity with Λ 0 < Hypergravity
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
28 / 28 The whole discussion can be pursued purely algebraically, without invoking geometrical concepts. The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal.
Charges and asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
28 / 28 without invoking geometrical concepts. The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal. Charges and The whole discussion can be pursued purely algebraically, asymptotic analysis
Hypersymmetry bounds
Black holes
Conclusions
28 / 28 The question remains, however : can one define an invariant geometry in the presence of higher spin gauge fields ? Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal. Charges and The whole discussion can be pursued purely algebraically, asymptotic analysis without invoking geometrical concepts. Hypersymmetry bounds
Black holes
Conclusions
28 / 28 Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal. Charges and The whole discussion can be pursued purely algebraically, asymptotic analysis without invoking geometrical concepts. Hypersymmetry bounds The question remains, however : can one define an invariant Black holes geometry in the presence of higher spin gauge fields ? Conclusions
28 / 28 One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal. Charges and The whole discussion can be pursued purely algebraically, asymptotic analysis without invoking geometrical concepts. Hypersymmetry bounds The question remains, however : can one define an invariant Black holes geometry in the presence of higher spin gauge fields ? Conclusions Another question is : can we account for all the bounds ?
28 / 28 Conclusions and comments
Hypersymmetric black holes in 2+1 gravity Marc Henneaux Hypersymmetry bounds exist, are non trivial and are interesting. Introduction They provide nonlinear constraints on the bosonic charges. Three- dimensional pure In the case of 2+1 hypergravity, the black holes that saturate the gravity with Λ 0 < Hypergravity bounds are extremal. Charges and The whole discussion can be pursued purely algebraically, asymptotic analysis without invoking geometrical concepts. Hypersymmetry bounds The question remains, however : can one define an invariant Black holes geometry in the presence of higher spin gauge fields ? Conclusions Another question is : can we account for all the bounds ? One should consider more complete models that include supersymmetry, higher spin hypersymmetry. Perhaps one must go all the way to an infinite number of spins...
28 / 28