[PPT] - Hypersymmetric black holes in 2+1 gravity Hypergravity Charges and PowerPoint Presentation

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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

Hypersymmetric black holes in 2+1 gravity

Marc Henneaux ETH, November 2015

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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

Introduction

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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

Introduction

Supersymmetry is well known to have deep consequences.

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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

Introduction

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

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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

Introduction

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 ?

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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

Introduction

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 ?

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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

Introduction

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.

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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

Introduction

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)

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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

Introduction

I will successively discuss :

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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

Introduction

I will successively discuss : Anti-de Sitter Einstein gravity in three spacetime dimensions

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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

Introduction

I will successively discuss : Anti-de Sitter Einstein gravity in three spacetime dimensions Anti-de Sitter Hypergravity in three spacetime dimensions

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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

Introduction

I will successively discuss : Anti-de Sitter Einstein gravity in three spacetime dimensions Anti-de Sitter Hypergravity in three spacetime dimensions Charges and asymptotic analysis

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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

Introduction

I will successively discuss : 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

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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

Introduction

I will successively discuss : 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

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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

Anti-de Sitter group in three dimensions

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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

Anti-de Sitter group in three dimensions

The AdS algebra in D dimensions is so(D−1,2)

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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

Anti-de Sitter group in three dimensions

The AdS algebra in D dimensions is so(D−1,2) In three dimensions, this gives so(2,2).

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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

Anti-de Sitter group in three dimensions

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)

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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

Anti-de Sitter group in three dimensions

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)

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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

Anti-de Sitter group in three dimensions

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).

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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

Anti-de Sitter group in three dimensions

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

f isomorphisms so(2,1) ≃ sl(2,R) ≃ sp(2,R) ≃ su(1,1)

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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

Chern-Simons reformulation

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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

Chern-Simons reformulation

AdS gravity can be reformulated as an sl(2,R)⊕sl(2,R) Chern-Simons theory.

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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

Chern-Simons reformulation

AdS gravity can be reformulated as an sl(2,R)⊕sl(2,R) Chern-Simons theory. The action reads I[A+,A−] = ICS[A+]−ICS[A−] where A+, A− are connections taking values in the algebra sl(2,R),

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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

Chern-Simons reformulation

AdS gravity can be reformulated as an sl(2,R)⊕sl(2,R) Chern-Simons theory. The action reads I[A+,A−] = ICS[A+]−ICS[A−] where A+, A− are connections taking values in the algebra sl(2,R), and where ICS[A] is the Chern-Simons action ICS[A] = k 4π

M

Tr

A∧dA+ 2

3A∧A∧A

.

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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

Chern-Simons reformulation

AdS gravity can be reformulated as an sl(2,R)⊕sl(2,R) Chern-Simons theory. The action reads I[A+,A−] = ICS[A+]−ICS[A−] where A+, A− are connections taking values in the algebra sl(2,R), and where ICS[A] is the Chern-Simons action ICS[A] = k 4π

M

Tr

A∧dA+ 2

3A∧A∧A

.

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

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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

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

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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

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

The relationship between the sl(2,R) connections A+, A− and the gravitational variables (dreibein and spin connection) is

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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

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

The relationship between the sl(2,R) connections A+, A− and the gravitational variables (dreibein and spin connection) is A+a

µ = ωa µ + 1

ℓea

µ

and A−a

µ = ωa µ − 1

ℓea

µ,

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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

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

The relationship between the sl(2,R) connections A+, A− and the gravitational variables (dreibein and spin connection) is A+a

µ = ωa µ + 1

ℓea

µ

and A−a

µ = ωa µ − 1

ℓea

µ,

in terms of which one finds indeed I[e,ω] = 1 8πG

M

d3x 1 2eR+ e ℓ2

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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

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

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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

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

The absence of local degrees of freedom manifests itself in the Chern-Simons formulation through the fact that the connection is flat,

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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

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

The absence of local degrees of freedom manifests itself in the Chern-Simons formulation through the fact that the connection is flat, F = 0,

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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

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

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.

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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

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

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.

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

The Chern-Simons formulation is very convenient because it allows for generalizations.

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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.

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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".

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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.)

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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".

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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).

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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

D = 3 Pure N-extended Supergravities as Chern-Simons theories

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

nly bosonic fields of “spins" 2 and 1 (and a single “graviton").

The second condition ensures that spinors are spin- 3

2 fields.

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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

Higher spin gauge theories

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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

Higher spin gauge theories

But one may relax these conditions !

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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

Higher spin gauge theories

But one may relax these conditions ! This leads to higher spin gauge theories in 3D.

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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

Higher spin gauge theories

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.

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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

Higher spin gauge theories

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.

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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

Higher spin gauge theories

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.

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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

Hypergravity

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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

Hypergravity

The case of interest to us is obtained by replacing sl(2,R) by

sp(1,4).

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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

Hypergravity

The case of interest to us is obtained by replacing sl(2,R) by

sp(1,4).

More precisely, one replaces the gauge algebra sl(2,R)⊕sl(2,R) by

sp(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).

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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

Hypergravity

The case of interest to us is obtained by replacing sl(2,R) by

sp(1,4).

More precisely, one replaces the gauge algebra sl(2,R)⊕sl(2,R) by

sp(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 ?

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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

Hypergravity

The case of interest to us is obtained by replacing sl(2,R) by

sp(1,4).

More precisely, one replaces the gauge algebra sl(2,R)⊕sl(2,R) by

sp(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 spin-2 field, one spin-4 field and one spin- 5

2 field.

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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

Hypergravity

The case of interest to us is obtained by replacing sl(2,R) by

sp(1,4).

More precisely, one replaces the gauge algebra sl(2,R)⊕sl(2,R) by

sp(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 spin-2 field, one spin-4 field and one spin- 5

2 field.

The spin-4 field decouples in the limit of zero cosmological constant, where one gets the theory of Aragone and Deser (1984).

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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

Some conventions

Basis of osp(1|4) :

Li,Lj
=
i−j
Li+j ,
Li,Um
= (3i−m)Ui+m ,
Li,Sp
=

3 2 i−p

Si+p ,

[Um,Un] = 1 223 (m−n)

m2 +n2 −4
m2 +n2 − 2

3 mn−9

− 2

3 (mn−6)mn

Lm+n

+ 1 6 (m−n)

m2 −mn+n2 −7
Um+n ,
Um,Sp
=

1 233

2m3 −8m2p+20mp2 +82p−23m−40p3

Si+p ,

Sp,Sq
= Up+q +

1 223

6p2 −8pq+6q2 −9
Lp+q .

Here Li, with i = 0,±1, stand for the spin-2 generators that span the gravitational sl(2,R) subalgebra, while Um and Sp, with m = 0,±1,±2,±3 and p = ± 1

2,± 3 2, correspond to the spin-4 and

fermionic spin- 5

2 generators, respectively.

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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

Dynamics

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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

Dynamics

The action is I = ICS

A+

−ICS [A−]

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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

Dynamics

The action is I = ICS

A+

−ICS [A−] with ICS [A] = k4 4π

str
AdA+ 2

3A3

.

Here, the level, k4 = k/10, is expressed in terms of the Newton constant and the AdS radius according to k = ℓ/4G.

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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

Dynamics

The action is I = ICS

A+

−ICS [A−] with ICS [A] = k4 4π

str
AdA+ 2

3A3

.

Here, the level, k4 = k/10, is expressed in terms of the Newton constant and the AdS radius according to k = ℓ/4G. str [···] stands for the supertrace of the fundamental (5×5) matrix representation.

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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

Dynamics

The action is I = ICS

A+

−ICS [A−] with ICS [A] = k4 4π

str
AdA+ 2

3A3

.

Here, the level, k4 = k/10, is expressed in terms of the Newton constant and the AdS radius according to k = ℓ/4G. str [···] stands for the supertrace of the fundamental (5×5) matrix representation. The connection reads A+ = Ai

µLi +Bm µ Um +ψp µSp

and a similar expression holds for A−.

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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

Dynamics

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)),

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)), where eµ ∼ A+

µ −A− µ,

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)), where eµ ∼ A+

µ −A− µ,

and the spin- 5

2 field is ψµa, γaψµa = 0 (ψα µa ∼ ψp µ).

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)), where eµ ∼ A+

µ −A− µ,

and the spin- 5

2 field is ψµa, γaψµa = 0 (ψα µa ∼ ψp µ).

The action is S[gµν,hµνρσ,ψµa] = SE +SF +S

5 2 +SI

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)), where eµ ∼ A+

µ −A− µ,

and the spin- 5

2 field is ψµa, γaψµa = 0 (ψα µa ∼ ψp µ).

The action is S[gµν,hµνρσ,ψµa] = SE +SF +S

5 2 +SI

where SE is the Einstein action, SF the (covariantized) Fronsdal action for a spin-4 field, S

5 2 the (covariantized) spin- 5

2 action and

SI stands for the higher order interaction terms necessary to make the theory consistent.

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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

Dynamics

In terms of the two osp(1|4) connections A+ and A−, the metric and spin-4 field are defined by gµν ∼ str(eµeν) , hµνρσ ∼ str(eµeνeρeσ)+astr(e(µeν)str(eρeσ)), where eµ ∼ A+

µ −A− µ,

and the spin- 5

2 field is ψµa, γaψµa = 0 (ψα µa ∼ ψp µ).

The action is S[gµν,hµνρσ,ψµa] = SE +SF +S

5 2 +SI

where SE is the Einstein action, SF the (covariantized) Fronsdal action for a spin-4 field, S

5 2 the (covariantized) spin- 5

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.

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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

Absence of a well-defined geometry

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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

Absence of a well-defined geometry

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.

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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

Absence of a well-defined geometry

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.

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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

Absence of a well-defined geometry

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.

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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

Absence of a well-defined geometry

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.

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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

Boundary conditions - Pure gravity

We first consider pure gravity

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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

Boundary conditions - Pure gravity

We first consider pure gravity 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.

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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

Boundary conditions - Pure gravity

We first consider pure gravity 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.

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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

Boundary conditions - Pure gravity

We first consider pure gravity 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

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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

Boundary conditions - Pure gravity

We first consider pure gravity 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 A±

ϕ

r,ϕ
−

→

r→∞ L±1 − 2π

k L ± ϕ

L∓1 +O

1 r

,

and A±

r −

→

r→∞ O

1 r

.

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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

Boundary conditions - Pure gravity

We first consider pure gravity 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 A±

ϕ

r,ϕ
−

→

r→∞ L±1 − 2π

k L ± ϕ

L∓1 +O

1 r

,

and A±

r −

→

r→∞ O

1 r

.

Coussaert, Henneaux, van Driel 1995

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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

Asymptotic symmetries

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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

Asymptotic symmetries

The “asymptotic symmetries" are those gauge transformations δA±

i = ∂iΛ± +[A± i ,Λ±] that preserve the boundary conditions, i.e.,

such that A±

i +δA± i fulfills also the boundary conditions.

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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

Asymptotic symmetries

The “asymptotic symmetries" are those gauge transformations δA±

i = ∂iΛ± +[A± i ,Λ±] that preserve the boundary conditions, i.e.,

such that A±

i +δA± i fulfills also the boundary conditions.

They are given by Λ± − →

r→∞

±ǫ±

ϕ
L±1 − 2π

k L ± ϕ

L∓1
∓ǫ′

±

ϕ
L0 ± 1

2ǫ′′

±

ϕ
L∓1

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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

Asymptotic symmetries

The “asymptotic symmetries" are those gauge transformations δA±

i = ∂iΛ± +[A± i ,Λ±] that preserve the boundary conditions, i.e.,

such that A±

i +δA± i fulfills also the boundary conditions.

They are given by Λ± − →

r→∞

±ǫ±

ϕ
L±1 − 2π

k L ± ϕ

L∓1
∓ǫ′

±

ϕ
L0 ± 1

2ǫ′′

±

ϕ
L∓1

The functions ǫ±

ϕ
are arbitrary functions of ϕ and parametrize

the asymptotic symmetries.

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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

Generators of asymptotic symmetries - Virasoro algebra

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

These generators obey the Virasoro algebra.

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

These generators obey the Virasoro algebra. More precisely, the Fourier components L ±

n obey, in terms of the

Poisson bracket, the Virasoro algebra with the classical central charge c = 6k = 3ℓ/2G,

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

These generators obey the Virasoro algebra. More precisely, the Fourier components L ±

n obey, in terms of the

Poisson bracket, the Virasoro algebra with the classical central charge c = 6k = 3ℓ/2G, i

L ±

m,L ± n

PB = (m−n)L ±

m+n + k

2m3δm+n,0.

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

These generators obey the Virasoro algebra. More precisely, the Fourier components L ±

n obey, in terms of the

Poisson bracket, the Virasoro algebra with the classical central charge c = 6k = 3ℓ/2G, i

L ±

m,L ± n

PB = (m−n)L ±

m+n + k

2m3δm+n,0. and commute between themselves

L +

m,L − n

PB = 0 (2D

conformal algebra).

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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

Generators of asymptotic symmetries - Virasoro algebra

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 Q±[ǫ±] = ±

r→∞

ǫ±

ϕ
L ±

ϕ

dϕ

These generators obey the Virasoro algebra. More precisely, the Fourier components L ±

n obey, in terms of the

Poisson bracket, the Virasoro algebra with the classical central charge c = 6k = 3ℓ/2G, i

L ±

m,L ± n

PB = (m−n)L ±

m+n + k

2m3δm+n,0. and commute between themselves

L +

m,L − n

PB = 0 (2D

conformal algebra). Thus, the Virasoro algebra emerges in the reduction procedure enforced by the AdS boundary conditions.

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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

Hypergravity

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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

Hypergravity

The same asymptotic analysis can be performed for hypergravity.

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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

Hypergravity

The same asymptotic analysis can be performed for hypergravity. One gets an enhancement of the asymptotic algebra, from the Virasoro algebra to the W(2, 5

2 ,4)-superalgebra, which contains the

Virasoro generators but also generators of higher conformal weights.

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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

Hypergravity

The same asymptotic analysis can be performed for hypergravity. One gets an enhancement of the asymptotic algebra, from the Virasoro algebra to the W(2, 5

2 ,4)-superalgebra, which contains the

Virasoro generators but also generators of higher conformal weights. How does this proceed ?

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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

Hypergravity

The same asymptotic analysis can be performed for hypergravity. One gets an enhancement of the asymptotic algebra, from the Virasoro algebra to the W(2, 5

2 ,4)-superalgebra, which contains the

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)

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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

Boundary conditions

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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

Boundary conditions

The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type

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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

Boundary conditions

The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type A±

ϕ

r,ϕ
−

→

r→∞ L±1−2π

k L ± ϕ

L∓1+ π

5k U ± ϕ

U∓3−2π

k ψ± ϕ

S∓ 3

2 ,

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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

Boundary conditions

The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type A±

ϕ

r,ϕ
−

→

r→∞ L±1−2π

k L ± ϕ

L∓1+ π

5k U ± ϕ

U∓3−2π

k ψ± ϕ

S∓ 3

2 ,

Again, the non trivial fields L ± ϕ

, U ±

ϕ

and ψ±(ϕ) appear

along the lowest (highest)-weight generators.

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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

Boundary conditions

The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type A±

ϕ

r,ϕ
−

→

r→∞ L±1−2π

k L ± ϕ

L∓1+ π

5k U ± ϕ

U∓3−2π

k ψ± ϕ

S∓ 3

2 ,

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 Q± ǫ±,χ±,ϑ±

= ±
dϕ
ǫ±L ± +χ±U ± −iϑ±ψ±

,

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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

Boundary conditions

The asymptotic conditions that generalize those found for pure gravity are again of Drinfeld-Sokolov type A±

ϕ

r,ϕ
−

→

r→∞ L±1−2π

k L ± ϕ

L∓1+ π

5k U ± ϕ

U∓3−2π

k ψ± ϕ

S∓ 3

2 ,

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 Q± ǫ±,χ±,ϑ±

= ±
dϕ
ǫ±L ± +χ±U ± −iϑ±ψ±

, with ǫ±, χ± and ϑ± arbitrary functions of ϕ. The “charges" L ± ϕ

, U ±

ϕ

and ψ±(ϕ) form the W(2, 5

2 ,4)-superalgebra.

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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

W(2, 5

2,4)-superalgebra i[Lm,Ln]PB = (m−n)Lm+n + k 2 m3δ0

m+n ,

i[Lm,Un]PB = (3m−n)Um+n , i

Lm,ψn
PB =

3 2 m−n

ψm+n ,

i[Um,Un]PB = 1 2232 (m−n)

3m4 −2m3n+4m2n2 −2mn3 +3n4

Lm+n + 1 6 (m−n)

m2 −mn+n2

Um+n − 233π k (m−n)Λ(6)

m+n

− 72π 32k (m−n)

m2 +4mn+n2

Λ(4)

m+n +

k 2332 m7δ0

m+n ,

i

Um,ψn
PB =

1 223

m3 −4m2n+10mn2 −20n3

ψm+n − 23π 3k iΛ(11/2)

m+n

+ π 3k (23m−82n)Λ(9/2)

m+n ,

i

ψm,ψn
PB = Um+n + 1

2

m2 − 4

3 mn+n2

Lm+n + 3π

k Λ(4)

m+n + k

6 m4δ0

m+n ,

The generators Un, ψn have respective conformal weights 4 and 5

2 ;

unchanged central charge c = 6k = 3ℓ

2G (two copies).

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Hypersymmetry bounds

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Hypersymmetry bounds

The fermions can be anti-periodic or periodic.

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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

Hypersymmetry bounds

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").

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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

Hypersymmetry bounds

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

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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

Hypersymmetry bounds

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

(2π)−1 ˆ ψ0 ˆ ψ0 + ˆ ψ0 ˆ ψ0

= 2

2π ˆ ψ2

0 = U + 3π

k L 2 ≥ 0.

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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

Hypersymmetry bounds

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

(2π)−1 ˆ ψ0 ˆ ψ0 + ˆ ψ0 ˆ ψ0

= 2

2π ˆ ψ2

0 = U + 3π

k L 2 ≥ 0. (The quantum W (2,5/2,4)-superalgebra, with the unitarity conditions L†

m = L−m, U† m = U−m, ψ† m = ψ−m implied by the

classical reality conditions, admits arbitrarily large values of the central charge.)

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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

Hypersymmetry bounds

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

(2π)−1 ˆ ψ0 ˆ ψ0 + ˆ ψ0 ˆ ψ0

= 2

2π ˆ ψ2

0 = U + 3π

k L 2 ≥ 0. (The quantum W (2,5/2,4)-superalgebra, with the unitarity conditions L†

m = L−m, U† m = U−m, ψ† m = ψ−m implied by the

classical reality conditions, admits arbitrarily large values of the central charge.) This is a nonlinear bound.

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Black holes - Euclidean continuation

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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

Black holes - Euclidean continuation

In the absence of a well-defined geometry, black holes and black hole thermodynamics are defined through the Euclidean continuation.

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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

Black holes - Euclidean continuation

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)

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Black holes - Euclidean continuation

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.

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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

Black holes - Euclidean continuation

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)

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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

Black hole topology - Euclidean formulation

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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

Black hole topology - Euclidean formulation

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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

Black hole topology - Euclidean formulation

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.

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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

Black hole in Chern-Simons formulation - Definition

More precisely :

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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

Black hole in Chern-Simons formulation - Definition

More precisely : 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,

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SLIDE 121

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

Black hole in Chern-Simons formulation - Definition

More precisely : 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).

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SLIDE 122

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

Black hole in Chern-Simons formulation - Definition

More precisely : 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,

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SLIDE 123

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

Black hole in Chern-Simons formulation - Definition

More precisely : 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).

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SLIDE 124

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

Black hole in Chern-Simons formulation - Definition

More precisely : 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.

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SLIDE 125

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

Hypergravity black holes

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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

Hypergravity black holes

The hypergravity Euclidean black hole is then a flat

sp(1|4;C)-connection defined on the solid torus,

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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

Hypergravity black holes

The hypergravity Euclidean black hole is then a flat

sp(1|4;C)-connection defined on the solid torus,
beying the above boundary conditions

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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

Hypergravity black holes

The hypergravity Euclidean black hole is then a flat

sp(1|4;C)-connection defined on the solid torus,
beying the above boundary conditions

and regular everywhere, including at the origin r+.

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SLIDE 129

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

Hypergravity black holes

The hypergravity Euclidean black hole is then a flat

sp(1|4;C)-connection defined on the solid torus,
beying the above boundary conditions

and regular everywhere, including at the origin r+. This is the generalization of absence of conical singularity at the horizon.

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SLIDE 130

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

Hypergravity black holes

The hypergravity Euclidean black hole is then a flat

sp(1|4;C)-connection defined on the solid torus,
beying 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.

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SLIDE 131

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

Hypergravity black holes

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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

Hypergravity black holes

The Euclidean connection for the black hole is explicitly given by

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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

Hypergravity black holes

The Euclidean connection for the black hole is explicitly given by Aϕ = L1 − 2π k L L−1 + π 5k U U−3 ,

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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

Hypergravity black holes

The Euclidean connection for the black hole is explicitly given by Aϕ = L1 − 2π k L L−1 + π 5k U U−3 , where L and U are now complex constants (related to the Lorentzian L ± and U ±) (mass, angular momentum and spin-4 charges).

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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

Hypergravity black holes

The Euclidean connection for the black hole is explicitly given by Aϕ = L1 − 2π k L L−1 + π 5k U U−3 , 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

rigin) determines ξ and µ in terms of the charges L , U .

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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

Hypergravity black hole -Thermodynamics

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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

Hypergravity black hole -Thermodynamics

The Euclidean action gives the entropy

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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

Hypergravity black hole -Thermodynamics

The Euclidean action gives the entropy and the thermodynamics can be consistently defined

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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

Hypergravity black hole -Thermodynamics

The Euclidean action gives the entropy and the thermodynamics can be consistently defined provided the charges are within the “extremal limit"

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SLIDE 140

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

Hypergravity black hole -Thermodynamics

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.

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SLIDE 141

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

Hypergravity black hole -Thermodynamics

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.

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SLIDE 142

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

Hypergravity black hole -Thermodynamics

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. One finds as extremality bounds L ± ≥ 0, k

3πU ± ≤ 24 32

L ±2 and

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SLIDE 143

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

Hypergravity black hole -Thermodynamics

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. One finds as extremality bounds L ± ≥ 0, k

3πU ± ≤ 24 32

L ±2 and

−

L ±2 ≤ k

3πU ±

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SLIDE 144

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

Hypergravity black hole -Thermodynamics

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. One finds as extremality bounds L ± ≥ 0, k

3πU ± ≤ 24 32

L ±2 and

−

L ±2 ≤ k

3πU ± This last bound is just the hypersymmetric bound found above from the algebra.

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SLIDE 145

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

Hypergravity black hole -Thermodynamics

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. One finds as extremality bounds L ± ≥ 0, k

3πU ± ≤ 24 32

L ±2 and

−

L ±2 ≤ k

3πU ± 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).

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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

Conclusions and comments

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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

Conclusions and comments

Hypersymmetry bounds exist, are non trivial and are interesting.

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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

Conclusions and comments

Hypersymmetry bounds exist, are non trivial and are interesting. They provide nonlinear constraints on the bosonic charges.

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SLIDE 149

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

Conclusions and comments

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.

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SLIDE 150

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

Conclusions and comments

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,

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SLIDE 151

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

Conclusions and comments

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.

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SLIDE 152

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

Conclusions and comments

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 ?

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SLIDE 153

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

Conclusions and comments

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 ?

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SLIDE 154

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

Conclusions and comments

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...

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