Hamiltonian Formalism for Gauge Systems and Manifestly Duality Symmetric Actions Marc Henneaux

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction Hamiltonian formalism for gauge systems and

SO(2)-Electric- magnetic manifestly duality symmetric actions gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in Marc Henneaux D 4 = Twisted self-duality

Conclusions Quim Fest - September 2015

1 / 20 Quim was one of the ﬁrst physicists to recognize the beauty and power of the Hamiltonian formalism developed by Dirac for gauge systems, always focusing on physical applications. I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

2 / 20 always focusing on physical applications. I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries.

Introduction

Conclusions

2 / 20 I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries.

Introduction

Conclusions

2 / 20 and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the ﬁrst physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 = I will describe here an application to SO(2) electric-magnetic SO(2)-Electric- magnetic duality duality, for higher spins in D 4 = Twisted self-duality

Conclusions

2 / 20 It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries.

Introduction

Conclusions

2 / 20 and involves both gauge and rigid symmetries.

Introduction

2 / 20 Introduction

2 / 20 Following the work of Cremmer and Julia, it has been conjectured that the inﬁnite-dimensional Kac-Moody algebra E10

−1 0 1 2 3 4 5 6 7

or E11 11

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

3 / 20 it has been conjectured that the inﬁnite-dimensional Kac-Moody algebra E10

−1 0 1 2 3 4 5 6 7

or E11 11

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

3 / 20 8

−1 0 1 2 3 4 5 6 7

or E11 11

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the inﬁnite-dimensional Kac-Moody actions Marc Henneaux algebra E10

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

3 / 20 or E11 11

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

8 Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- −1 0 1 2 3 4 5 6 7 magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

3 / 20 11

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

8 Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- −1 0 1 2 3 4 5 6 7 magnetic duality for higher spins in or E D 4 11 = Twisted self-duality

Conclusions

3 / 20 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

Introduction

8 Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- −1 0 1 2 3 4 5 6 7 magnetic duality for higher spins in or E D 4 11 = 11 Twisted self-duality

Conclusions

10 9 8 7 6 5 4 3 2 1

3 / 20 Introduction

8 Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- −1 0 1 2 3 4 5 6 7 magnetic duality for higher spins in or E D 4 11 = 11 Twisted self-duality

Conclusions

10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.

3 / 20 One crucial feature of these algebras is that they have duality built in, i.e., they treat democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears in the spectrum of fields, its dual D p 2-form gauge field also − − appears. Similarly, the graviton and its dual, described by a field with Young symmetry

      D 3 boxes −     

simultaneously appear.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

4 / 20 Whenever a (dynamical) p-form gauge field appears in the spectrum of fields, its dual D p 2-form gauge field also − − appears. Similarly, the graviton and its dual, described by a field with Young symmetry

      D 3 boxes −     

simultaneously appear.

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

4 / 20 Similarly, the graviton and its dual, described by a ﬁeld with Young symmetry

      D 3 boxes −     

simultaneously appear.

Introduction

Hamiltonian formalism for gauge systems One crucial feature of these algebras is that they have duality and manifestly duality built in, i.e., they treat democratically all fields and their duals. symmetric actions Whenever a (dynamical) p-form gauge field appears in the Marc Henneaux spectrum of fields, its dual D p 2-form gauge field also − − Introduction appears. SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

4 / 20       D 3 boxes −     

simultaneously appear.

Introduction

Hamiltonian formalism for gauge systems One crucial feature of these algebras is that they have duality and manifestly duality built in, i.e., they treat democratically all fields and their duals. symmetric actions Whenever a (dynamical) p-form gauge field appears in the Marc Henneaux spectrum of fields, its dual D p 2-form gauge field also − − Introduction appears. SO(2)-Electric- magnetic Similarly, the graviton and its dual, described by a field with gravitational duality in D 4 Young symmetry = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

4 / 20 simultaneously appear.

Introduction

Hamiltonian formalism for gauge systems One crucial feature of these algebras is that they have duality and manifestly duality built in, i.e., they treat democratically all fields and their duals. symmetric actions Whenever a (dynamical) p-form gauge field appears in the Marc Henneaux spectrum of fields, its dual D p 2-form gauge field also − − Introduction appears. SO(2)-Electric- magnetic Similarly, the graviton and its dual, described by a field with gravitational duality in D 4 Young symmetry = SO(2)-Electric- magnetic duality for higher spins in  D 4  =  Twisted  self-duality   Conclusions D 3 boxes −     

4 / 20 Introduction

Hamiltonian formalism for gauge systems One crucial feature of these algebras is that they have duality and manifestly duality built in, i.e., they treat democratically all fields and their duals. symmetric actions Whenever a (dynamical) p-form gauge field appears in the Marc Henneaux spectrum of fields, its dual D p 2-form gauge field also − − Introduction appears. SO(2)-Electric- magnetic Similarly, the graviton and its dual, described by a field with gravitational duality in D 4 Young symmetry = SO(2)-Electric- magnetic duality for higher spins in  D 4  =  Twisted  self-duality   Conclusions D 3 boxes −     

simultaneously appear.

4 / 20 However, in spite of many efforts, proving the conjecture has not been achieved yet. One difﬁculty might lie in the fact that physically consistent action principles where duality is manifest have unconventional features.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

5 / 20 proving the conjecture has not been achieved yet. One difﬁculty might lie in the fact that physically consistent action principles where duality is manifest have unconventional features.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction However, in spite of many efforts, SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

5 / 20 One difﬁculty might lie in the fact that physically consistent action principles where duality is manifest have unconventional features.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction However, in spite of many efforts, SO(2)-Electric- magnetic gravitational proving the conjecture has not been achieved yet. duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

5 / 20 Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction However, in spite of many efforts, SO(2)-Electric- magnetic gravitational proving the conjecture has not been achieved yet. duality in D 4 = SO(2)-Electric- One difﬁculty might lie in the fact that physically consistent magnetic duality for higher spins in action principles where duality is manifest have unconventional D 4 = features. Twisted self-duality

Conclusions

5 / 20 Electric-magnetic gravitational duality in 4 spacetime dimensions ; Electric-magnetic duality for higher spins in 4 spacetime dimensions (very brieﬂy) ; Twisted self-duality (also very brieﬂy).

The purpose of this talk is to discuss manifestly duality-symmetric actions. We shall successively consider three cases :

Mix of review material and more recent results. We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

We shall successively consider three cases :

Mix of review material and more recent results. We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

6 / 20 Electric-magnetic gravitational duality in 4 spacetime dimensions ; Electric-magnetic duality for higher spins in 4 spacetime dimensions (very briefly) ; Twisted self-duality (also very briefly). Mix of review material and more recent results. We shall finally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

SO(2)-Electric- We shall successively consider three cases : magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

6 / 20 Electric-magnetic duality for higher spins in 4 spacetime dimensions (very briefly) ; Twisted self-duality (also very briefly). Mix of review material and more recent results. We shall finally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

SO(2)-Electric- We shall successively consider three cases : magnetic gravitational Electric-magnetic gravitational duality in 4 spacetime dimensions ; duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

6 / 20 Twisted self-duality (also very brieﬂy). Mix of review material and more recent results. We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

SO(2)-Electric- We shall successively consider three cases : magnetic gravitational Electric-magnetic gravitational duality in 4 spacetime dimensions ; duality in D 4 = Electric-magnetic duality for higher spins in 4 spacetime SO(2)-Electric- magnetic duality dimensions (very brieﬂy) ; for higher spins in D 4 = Twisted self-duality

Conclusions

6 / 20 Mix of review material and more recent results. We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

Conclusions

6 / 20 We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

Conclusions

6 / 20 Introduction

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The purpose of this talk is to discuss manifestly Marc Henneaux duality-symmetric actions. Introduction

Conclusions We shall ﬁnally conclude with comments on the potential emergence of the spacetime geometry from duality.

6 / 20 The Riemann tensor 1 ¡ ¢ R ∂ ∂ρhµσ ∂µ∂ρh ∂ ∂σhµρ ∂µ∂σh λµρσ = −2 λ − λσ − λ + λρ fulﬁlls the identity R 0. λ[µρσ] =

The Einstein equations are R 0. µν = This implies that the dual Riemann tensor

1 αβ ∗R ² R λµρσ = 2 λµαβ ρσ

also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

and conversely.

Duality invariance of the Einstein equations

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

7 / 20 The Einstein equations are R 0. µν = This implies that the dual Riemann tensor

1 αβ ∗R ² R λµρσ = 2 λµαβ ρσ

also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

and conversely.

Duality invariance of the Einstein equations

Hamiltonian formalism for The Riemann tensor gauge systems and manifestly duality 1 ¡ ¢ symmetric Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ actions = −2 − − + Marc Henneaux fulﬁlls the identity Introduction Rλ[µρσ] 0. SO(2)-Electric- = magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

7 / 20 This implies that the dual Riemann tensor

1 αβ ∗R ² R λµρσ = 2 λµαβ ρσ

also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

and conversely.

Duality invariance of the Einstein equations

Hamiltonian formalism for The Riemann tensor gauge systems and manifestly duality 1 ¡ ¢ symmetric Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ actions = −2 − − + Marc Henneaux fulﬁlls the identity Introduction Rλ[µρσ] 0. SO(2)-Electric- = magnetic gravitational duality in D 4 = The Einstein equations are Rµν 0. SO(2)-Electric- = magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

7 / 20 also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

and conversely.

Duality invariance of the Einstein equations

Hamiltonian formalism for The Riemann tensor gauge systems and manifestly duality 1 ¡ ¢ symmetric Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ actions = −2 − − + Marc Henneaux fulﬁlls the identity Introduction Rλ[µρσ] 0. SO(2)-Electric- = magnetic gravitational duality in D 4 = The Einstein equations are Rµν 0. SO(2)-Electric- = magnetic duality This implies that the dual Riemann tensor for higher spins in D 4 = 1 αβ Twisted ∗Rλµρσ ²λµαβR ρσ self-duality = 2 Conclusions

7 / 20 and conversely.

Duality invariance of the Einstein equations

Hamiltonian formalism for The Riemann tensor gauge systems and manifestly duality 1 ¡ ¢ symmetric Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ actions = −2 − − + Marc Henneaux fulﬁlls the identity Introduction Rλ[µρσ] 0. SO(2)-Electric- = magnetic gravitational duality in D 4 = The Einstein equations are Rµν 0. SO(2)-Electric- = magnetic duality This implies that the dual Riemann tensor for higher spins in D 4 = 1 αβ Twisted ∗Rλµρσ ²λµαβR ρσ self-duality = 2 Conclusions

also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

7 / 20 Duality invariance of the Einstein equations

Hamiltonian formalism for The Riemann tensor gauge systems and manifestly duality 1 ¡ ¢ symmetric Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ actions = −2 − − + Marc Henneaux fulﬁlls the identity Introduction Rλ[µρσ] 0. SO(2)-Electric- = magnetic gravitational duality in D 4 = The Einstein equations are Rµν 0. SO(2)-Electric- = magnetic duality This implies that the dual Riemann tensor for higher spins in D 4 = 1 αβ Twisted ∗Rλµρσ ²λµαβR ρσ self-duality = 2 Conclusions

also fulﬁlls ∗R 0, ∗R 0 λ[µρσ] = µν =

and conversely.

7 / 20 It follows that the Einstein equations are invariant under the duality rotations

R cosαR sinα ∗R → − ∗R sinαR cosα ∗R, → +

or in (3 1)- fashion, +

E ij cosαE ij sinαBij → − Bij sinαE ij cosαBij → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

8 / 20 R cosαR sinα ∗R → − ∗R sinαR cosα ∗R, → +

or in (3 1)- fashion, +

E ij cosαE ij sinαBij → − Bij sinαE ij cosαBij → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

8 / 20 or in (3 1)- fashion, +

E ij cosαE ij sinαBij → − Bij sinαE ij cosαBij → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux R cosαR sinα ∗R → − Introduction ∗R sinαR cosα ∗R, SO(2)-Electric- → + magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

8 / 20 E ij cosαE ij sinαBij → − Bij sinαE ij cosαBij → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux R cosαR sinα ∗R → − Introduction ∗R sinαR cosα ∗R, SO(2)-Electric- → + magnetic gravitational duality in D 4 = or in (3 1)- fashion, SO(2)-Electric- + magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

8 / 20 where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux R cosαR sinα ∗R → − Introduction ∗R sinαR cosα ∗R, SO(2)-Electric- → + magnetic gravitational duality in D 4 = or in (3 1)- fashion, SO(2)-Electric- + magnetic duality for higher spins in D 4 E ij cosαE ij sinαBij = → − Twisted ij ij ij self-duality B sinαE cosαB Conclusions → +

8 / 20 This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux R cosαR sinα ∗R → − Introduction ∗R sinαR cosα ∗R, SO(2)-Electric- → + magnetic gravitational duality in D 4 = or in (3 1)- fashion, SO(2)-Electric- + magnetic duality for higher spins in D 4 E ij cosαE ij sinαBij = → − Twisted ij ij ij self-duality B sinαE cosαB Conclusions → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively.

8 / 20 Duality invariance of the Einstein equations

Hamiltonian formalism for It follows that the Einstein equations are invariant under the gauge systems and manifestly duality rotations duality symmetric actions Marc Henneaux R cosαR sinα ∗R → − Introduction ∗R sinαR cosα ∗R, SO(2)-Electric- → + magnetic gravitational duality in D 4 = or in (3 1)- fashion, SO(2)-Electric- + magnetic duality for higher spins in D 4 E ij cosαE ij sinαBij = → − Twisted ij ij ij self-duality B sinαE cosαB Conclusions → +

where E ij and Bij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.

8 / 20 Is this also a symmetry of the Pauli-Fierz action ?

S[hµν] =Z 1 4 £ ρ µν µν ρ µ ν µ ¤ d x ∂ h ∂ρhµν 2∂µh ∂ρh 2∂ h∂ hµν ∂ h∂µh . − 4 − ν + −

Note that the action is not expressed in terms of the curvature. The answer to the question turns out to be positive, just as for Maxwell’s theory, which was considered long ago in the pioneering work of Deser and Teitelboim (1976). To exhibit the SO(2) duality of the action, we follow the same steps as Deser and Teitelboim : Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

9 / 20 Note that the action is not expressed in terms of the curvature. The answer to the question turns out to be positive, just as for Maxwell’s theory, which was considered long ago in the pioneering work of Deser and Teitelboim (1976). To exhibit the SO(2) duality of the action, we follow the same steps as Deser and Teitelboim : Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

Conclusions

9 / 20 The answer to the question turns out to be positive, just as for Maxwell’s theory, which was considered long ago in the pioneering work of Deser and Teitelboim (1976). To exhibit the SO(2) duality of the action, we follow the same steps as Deser and Teitelboim : Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

Conclusions

9 / 20 which was considered long ago in the pioneering work of Deser and Teitelboim (1976). To exhibit the SO(2) duality of the action, we follow the same steps as Deser and Teitelboim : Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

Conclusions

9 / 20 To exhibit the SO(2) duality of the action, we follow the same steps as Deser and Teitelboim : Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly Is this also a symmetry of the Pauli-Fierz action ? duality symmetric actions S[hµν] Marc Henneaux =Z 1 4 £ ρ µν µν ρ µ ν µ ¤ Introduction d x ∂ h ∂ρhµν 2∂µh ∂ρh 2∂ h∂ hµν ∂ h∂µh . − 4 − ν + − SO(2)-Electric- magnetic gravitational duality in D 4 Note that the action is not expressed in terms of the curvature. = SO(2)-Electric- The answer to the question turns out to be positive, just as for magnetic duality for higher spins in Maxwell’s theory, D 4 = Twisted which was considered long ago in the pioneering work of Deser self-duality and Teitelboim (1976). Conclusions

9 / 20 Go to the ﬁrst-order (Hamiltonian) action and solve the constraints.

Duality invariance of the Pauli-Fierz action

9 / 20 Duality invariance of the Pauli-Fierz action

9 / 20 The dynamical variables of the Hamiltonian formalism are the components hij of the spatial metric and their conjugate momenta πij.

The temporal components h0µ are Lagrange multipliers for the constraints : H G[h] 0 (“Hamiltonian constraint"), ≡ = which expresses that the Einstein tensor Gij[h] of the spatial metric is traceless ( R[h] 0) ; ⇔ = and H j 2∂ πij 0 (“momentum constraint"), ≡ − i = which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

10 / 20 The temporal components h0µ are Lagrange multipliers for the constraints : H G[h] 0 (“Hamiltonian constraint"), ≡ = which expresses that the Einstein tensor Gij[h] of the spatial metric is traceless ( R[h] 0) ; ⇔ = and H j 2∂ πij 0 (“momentum constraint"), ≡ − i = which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

Conclusions

10 / 20 H G[h] 0 (“Hamiltonian constraint"), ≡ = which expresses that the Einstein tensor Gij[h] of the spatial metric is traceless ( R[h] 0) ; ⇔ = and H j 2∂ πij 0 (“momentum constraint"), ≡ − i = which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

Conclusions

10 / 20 which expresses that the Einstein tensor Gij[h] of the spatial metric is traceless ( R[h] 0) ; ⇔ = and H j 2∂ πij 0 (“momentum constraint"), ≡ − i = which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

Conclusions

10 / 20 and H j 2∂ πij 0 (“momentum constraint"), ≡ − i = which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions The dynamical variables of the Hamiltonian formalism are the Marc Henneaux components hij of the spatial metric and their conjugate Introduction momenta πij. SO(2)-Electric- magnetic The temporal components h0µ are Lagrange multipliers for the gravitational duality in D 4 = constraints : SO(2)-Electric- magnetic duality H G[h] 0 (“Hamiltonian constraint"), for higher spins in ≡ = D 4 ij = which expresses that the Einstein tensor G [h] of the spatial Twisted metric is traceless ( R[h] 0) ; self-duality ⇔ = Conclusions

10 / 20 which expresses that πij is transverse.

Duality invariance of the Pauli-Fierz action

10 / 20 Duality invariance of the Pauli-Fierz action

10 / 20 The explicit resolution of the constraints introduces two “prepotentials", one for the metric and one for its conjugate momentum. For instance, the momentum constraint ∂ πij 0 is solved by i = πij ²ipq²jrs∂ ∂ Z1 . = p r qs This is because any transverse symmetric tensor can be written ij 1 1 as the Einstein tensor G [Z ] for some symmetric prepotential Zij (Young symmetry type ). 1 The prepotential Zij has the gauge invariance

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij ... of a massless, conformal spin-2 ﬁeld !

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

11 / 20 For instance, the momentum constraint ∂ πij 0 is solved by i = πij ²ipq²jrs∂ ∂ Z1 . = p r qs This is because any transverse symmetric tensor can be written ij 1 1 as the Einstein tensor G [Z ] for some symmetric prepotential Zij (Young symmetry type ). 1 The prepotential Zij has the gauge invariance

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij ... of a massless, conformal spin-2 ﬁeld !

Duality invariance of the Pauli-Fierz action

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

11 / 20 This is because any transverse symmetric tensor can be written ij 1 1 as the Einstein tensor G [Z ] for some symmetric prepotential Zij (Young symmetry type ). 1 The prepotential Zij has the gauge invariance

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij ... of a massless, conformal spin-2 ﬁeld !

Duality invariance of the Pauli-Fierz action

Conclusions

11 / 20 1 The prepotential Zij has the gauge invariance

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij ... of a massless, conformal spin-2 ﬁeld !

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly The explicit resolution of the constraints introduces two duality symmetric “prepotentials", one for the metric and one for its conjugate actions Marc Henneaux momentum. For instance, the momentum constraint ∂ πij 0 is solved by Introduction i = SO(2)-Electric- magnetic ij ipq jrs 1 gravitational π ² ² ∂p∂rZqs. duality in D 4 = = SO(2)-Electric- This is because any transverse symmetric tensor can be written magnetic duality for higher spins in as the Einstein tensor Gij[Z1] for some symmetric prepotential Z1 D 4 ij = Twisted (Young symmetry type ). self-duality

Conclusions

11 / 20 ... of a massless, conformal spin-2 ﬁeld !

Duality invariance of the Pauli-Fierz action

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij

11 / 20 Duality invariance of the Pauli-Fierz action

δZ1 ∂ ξ1 ∂ ξ1 2²1δ ij = i j + j i + ij ... of a massless, conformal spin-2 ﬁeld !

11 / 20 Similarly, the solution of the Hamiltonian constraint for the 2 metric hij leads to a second prepotential Zij, which is also symmetric (Young symmetry type ). Indeed, the Hamiltonian constraint states that the Einstein tensor, which is transverse, should also be traceless Gij[h]δ 0. ij = This implies that Gij[h] is equal to the Cotton tensor Dij[Z2] of 2 some symmetric ﬁeld Zij, which is the second prepotential. 2 The Cotton tensor involves three derivatives of Zij, is symmetric, transverse and traceless, and also invariant under diffeomorphisms and Weyl rescalings

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

12 / 20 Indeed, the Hamiltonian constraint states that the Einstein tensor, which is transverse, should also be traceless Gij[h]δ 0. ij = This implies that Gij[h] is equal to the Cotton tensor Dij[Z2] of 2 some symmetric ﬁeld Zij, which is the second prepotential. 2 The Cotton tensor involves three derivatives of Zij, is symmetric, transverse and traceless, and also invariant under diffeomorphisms and Weyl rescalings

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

Duality invariance of the Pauli-Fierz action

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

12 / 20 This implies that Gij[h] is equal to the Cotton tensor Dij[Z2] of 2 some symmetric ﬁeld Zij, which is the second prepotential. 2 The Cotton tensor involves three derivatives of Zij, is symmetric, transverse and traceless, and also invariant under diffeomorphisms and Weyl rescalings

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

Duality invariance of the Pauli-Fierz action

Conclusions

12 / 20 2 The Cotton tensor involves three derivatives of Zij, is symmetric, transverse and traceless, and also invariant under diffeomorphisms and Weyl rescalings

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly Similarly, the solution of the Hamiltonian constraint for the duality 2 symmetric metric hij leads to a second prepotential Z , which is also actions ij Marc Henneaux symmetric (Young symmetry type ). Introduction Indeed, the Hamiltonian constraint states that the Einstein SO(2)-Electric- tensor, which is transverse, should also be traceless Gij[h]δ 0. magnetic ij = gravitational ij ij 2 duality in D 4 This implies that G [h] is equal to the Cotton tensor D [Z ] of = 2 SO(2)-Electric- some symmetric ﬁeld Z , which is the second prepotential. magnetic duality ij for higher spins in D 4 = Twisted self-duality

Conclusions

12 / 20 and also invariant under diffeomorphisms and Weyl rescalings

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

Duality invariance of the Pauli-Fierz action

Conclusions

12 / 20 Duality invariance of the Pauli-Fierz action

δZ2 ∂ ξ2 ∂ ξ2 2²2δ ij = i j + j i + ij

12 / 20 In terms of the prepotentials, the action takes the remarkable simple form

Z · Z ¸ ij S[Za ] dt 2 d3x²abD Z˙ H mn = − a bij −

where D ij D ij[Z ] is the Cotton tensor of the prepotential Z , a ≡ a aij and where the Hamiltonian is given by

Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a a Here, Rij is the Ricci tensor of the prepotential Zij. The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is also immediate.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

13 / 20 Z · Z ¸ ij S[Za ] dt 2 d3x²abD Z˙ H mn = − a bij −

where D ij D ij[Z ] is the Cotton tensor of the prepotential Z , a ≡ a aij and where the Hamiltonian is given by

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

13 / 20 where D ij D ij[Z ] is the Cotton tensor of the prepotential Z , a ≡ a aij and where the Hamiltonian is given by

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

13 / 20 and where the Hamiltonian is given by

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic ij ij gravitational where Da D [Za] is the Cotton tensor of the prepotential Zaij, duality in D 4 ≡ = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

13 / 20 Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a a Here, Rij is the Ricci tensor of the prepotential Zij. The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is also immediate.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic ij ij gravitational where Da D [Za] is the Cotton tensor of the prepotential Zaij, duality in D 4 ≡ = and where the Hamiltonian is given by SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

13 / 20 The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is also immediate.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic ij ij gravitational where Da D [Za] is the Cotton tensor of the prepotential Zaij, duality in D 4 ≡ = and where the Hamiltonian is given by SO(2)-Electric- magnetic duality for higher spins in D 4 Z µ 3 ¶ = H d3x 4Ra Rbij RaRb δ . Twisted ij ab self-duality = − 2

Conclusions a a Here, Rij is the Ricci tensor of the prepotential Zij.

13 / 20 Invariance under the gauge symmetries of the prepotentials is also immediate.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic ij ij gravitational where Da D [Za] is the Cotton tensor of the prepotential Zaij, duality in D 4 ≡ = and where the Hamiltonian is given by SO(2)-Electric- magnetic duality for higher spins in D 4 Z µ 3 ¶ = H d3x 4Ra Rbij RaRb δ . Twisted ij ab self-duality = − 2

Conclusions a a Here, Rij is the Ricci tensor of the prepotential Zij. The action is manifestly invariant under duality rotations of the prepotentials.

13 / 20 Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems In terms of the prepotentials, the action takes the remarkable and manifestly duality simple form symmetric actions Marc Henneaux Z · Z ¸ a 3 ab ij ˙ S[Z mn] dt 2 d x² Da Zbij H Introduction = − − SO(2)-Electric- magnetic ij ij gravitational where Da D [Za] is the Cotton tensor of the prepotential Zaij, duality in D 4 ≡ = and where the Hamiltonian is given by SO(2)-Electric- magnetic duality for higher spins in D 4 Z µ 3 ¶ = H d3x 4Ra Rbij RaRb δ . Twisted ij ab self-duality = − 2

Conclusions a a Here, Rij is the Ricci tensor of the prepotential Zij. The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is also immediate.

13 / 20 The emergence of the gauge symmetries of conformal gravity

δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij is somewhat intriguing. Henneaux-Teitelboim 2005 ; Bunster-Henneaux-Hörtner 2013 Just as for the Maxwell theory (Deser-Teitelboim), there is a tension between manifest duality invariance and manifest space-time covariance, since one loses manifest spacetime covariance. The theory has two spatial metrics. The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

14 / 20 Henneaux-Teitelboim 2005 ; Bunster-Henneaux-Hörtner 2013 Just as for the Maxwell theory (Deser-Teitelboim), there is a tension between manifest duality invariance and manifest space-time covariance, since one loses manifest spacetime covariance. The theory has two spatial metrics. The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

Duality invariance of the Pauli-Fierz action

Hamiltonian formalism for gauge systems and manifestly The emergence of the gauge symmetries of conformal gravity duality symmetric actions a a a a δZ ij ∂iξ j ∂jξ i 2² δij Marc Henneaux = + +

Introduction is somewhat intriguing. SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

14 / 20 Just as for the Maxwell theory (Deser-Teitelboim), there is a tension between manifest duality invariance and manifest space-time covariance, since one loses manifest spacetime covariance. The theory has two spatial metrics. The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

Duality invariance of the Pauli-Fierz action

Introduction is somewhat intriguing. SO(2)-Electric- magnetic Henneaux-Teitelboim 2005 ; Bunster-Henneaux-Hörtner 2013 gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

14 / 20 The theory has two spatial metrics. The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

Duality invariance of the Pauli-Fierz action

Introduction is somewhat intriguing. SO(2)-Electric- magnetic Henneaux-Teitelboim 2005 ; Bunster-Henneaux-Hörtner 2013 gravitational duality in D 4 = Just as for the Maxwell theory (Deser-Teitelboim), there is a SO(2)-Electric- magnetic duality tension between manifest duality invariance and manifest for higher spins in D 4 space-time covariance, since one loses manifest spacetime = Twisted covariance. self-duality

Conclusions

14 / 20 The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

Duality invariance of the Pauli-Fierz action

Conclusions The theory has two spatial metrics.

14 / 20 Duality invariance of the Pauli-Fierz action

Conclusions The theory has two spatial metrics. The extension to include interactions has not been sucessively worked out yet but there are positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological backgrounds : Julia-Levie-Ray 2005.

14 / 20 The analysis can be extended to higher spin s following the same lines. The solution of the constraint equations involves again two prepotentials, which are now symmetric tensors of rank s. The theory possesses (somewhat puzzingly) the gauge invariances of conformal higher spins, a a a δZ ∂(i ξ δ(i i λ i1 is 1 i2 is) 1 2 i3 is) ··· = ··· + ··· and the action can be written as in the spin-2 case, Z · Z ¸ a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

15 / 20 The solution of the constraint equations involves again two prepotentials, which are now symmetric tensors of rank s. The theory possesses (somewhat puzzingly) the gauge invariances of conformal higher spins, a a a δZ ∂(i ξ δ(i i λ i1 is 1 i2 is) 1 2 i3 is) ··· = ··· + ··· and the action can be written as in the spin-2 case, Z · Z ¸ a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

Hamiltonian formalism for gauge systems and manifestly The analysis can be extended to higher spin s following the same duality symmetric lines. actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

15 / 20 The theory possesses (somewhat puzzingly) the gauge invariances of conformal higher spins, a a a δZ ∂(i ξ δ(i i λ i1 is 1 i2 is) 1 2 i3 is) ··· = ··· + ··· and the action can be written as in the spin-2 case, Z · Z ¸ a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

Hamiltonian formalism for gauge systems and manifestly The analysis can be extended to higher spin s following the same duality symmetric lines. actions Marc Henneaux The solution of the constraint equations involves again two prepotentials, which are now symmetric tensors of rank s. Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

15 / 20 a a a δZ ∂(i ξ δ(i i λ i1 is 1 i2 is) 1 2 i3 is) ··· = ··· + ··· and the action can be written as in the spin-2 case, Z · Z ¸ a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

SO(2)-Electric- The theory possesses (somewhat puzzingly) the gauge magnetic gravitational invariances of conformal higher spins, duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

15 / 20 and the action can be written as in the spin-2 case, Z · Z ¸ a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

Conclusions

15 / 20 Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

Duality invariance of higher spin gauge ﬁelds

SO(2)-Electric- The theory possesses (somewhat puzzingly) the gauge magnetic gravitational invariances of conformal higher spins, duality in D 4 = a a a SO(2)-Electric- δZ ∂(i ξ δ(i i λ i1 is 1 i2 is) 1 2 i3 is) magnetic duality ··· = ··· + ··· for higher spins in and the action can be written as in the spin-2 case, D 4 = Twisted Z · Z ¸ self-duality a 3 ab i1 is S[Z ] dt 2 d x² B ··· Z˙ H i1 is a bi1 is Conclusions ··· = − ··· −

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ···

15 / 20 Duality invariance of higher spin gauge ﬁelds

i1 is a where B ··· is the Cotton tensor of Z . a i1 is ··· Bunster-Henneaux-Hörtner-Leonard 2014, Henneaux-Hörtner-Leonard 2015

15 / 20 In higher dimensions, for general p-forms or for the graviton, the curvature and its dual are tensors of different types. This is true for either electromagnetism or gravity. The duality-symmetric formulation is then based on the “(twisted) self-duality" reformulation of the theory, which involves in a symmetric way the original ﬁeld (p-form or graviton) and its dual. The duality-symmetric actions are two-potential actions that share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

Twisted self-duality

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

16 / 20 This is true for either electromagnetism or gravity. The duality-symmetric formulation is then based on the “(twisted) self-duality" reformulation of the theory, which involves in a symmetric way the original ﬁeld (p-form or graviton) and its dual. The duality-symmetric actions are two-potential actions that share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

Twisted self-duality

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

16 / 20 The duality-symmetric formulation is then based on the “(twisted) self-duality" reformulation of the theory, which involves in a symmetric way the original ﬁeld (p-form or graviton) and its dual. The duality-symmetric actions are two-potential actions that share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

Twisted self-duality

Conclusions

16 / 20 which involves in a symmetric way the original ﬁeld (p-form or graviton) and its dual. The duality-symmetric actions are two-potential actions that share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

Twisted self-duality

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions In higher dimensions, for general p-forms or for the graviton, the Marc Henneaux curvature and its dual are tensors of different types. Introduction This is true for either electromagnetism or gravity. SO(2)-Electric- magnetic The duality-symmetric formulation is then based on the gravitational duality in D 4 = “(twisted) self-duality" reformulation of the theory, SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

16 / 20 The duality-symmetric actions are two-potential actions that share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

Twisted self-duality

Conclusions

16 / 20 Twisted self-duality

Conclusions share features quite similar to the ones found above (Henneaux-Teitelboim 1988, Schwarz-Sen 1994, Hillmann 2010, Bunster-Henneaux 2011).

16 / 20 They can be obtained by following the same procedure (go Hamiltonian and introduce the second potential through solving the Gauss constraints). For instance, for a 1-form in ﬁve dimensions, the dual form B is a 2-form and the duality symmetric action reads

Z µ µ ¶ ¶ 0 4 1 ijkm 1 1 S[Ai,B ] dx d x ² H A˙ i FijB˙ H jk = 2 3! jkm − 4 km − with F dA, H dB and = = µ ¶ 1 1 jkm 1 ij H H H F Fij = 2 3! jkm + 2

Chern-Simons couplings can be included. Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

Twisted self-duality

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

17 / 20 For instance, for a 1-form in ﬁve dimensions, the dual form B is a 2-form and the duality symmetric action reads

Z µ µ ¶ ¶ 0 4 1 ijkm 1 1 S[Ai,B ] dx d x ² H A˙ i FijB˙ H jk = 2 3! jkm − 4 km − with F dA, H dB and = = µ ¶ 1 1 jkm 1 ij H H H F Fij = 2 3! jkm + 2

Chern-Simons couplings can be included. Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

Twisted self-duality

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

17 / 20 Z µ µ ¶ ¶ 0 4 1 ijkm 1 1 S[Ai,B ] dx d x ² H A˙ i FijB˙ H jk = 2 3! jkm − 4 km − with F dA, H dB and = = µ ¶ 1 1 jkm 1 ij H H H F Fij = 2 3! jkm + 2

Chern-Simons couplings can be included. Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

Twisted self-duality

Hamiltonian formalism for gauge systems They can be obtained by following the same procedure (go and manifestly duality Hamiltonian and introduce the second potential through solving symmetric actions the Gauss constraints). Marc Henneaux For instance, for a 1-form in ﬁve dimensions, the dual form B is a Introduction 2-form and the duality symmetric action reads SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

17 / 20 Chern-Simons couplings can be included. Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

Twisted self-duality

Hamiltonian formalism for gauge systems They can be obtained by following the same procedure (go and manifestly duality Hamiltonian and introduce the second potential through solving symmetric actions the Gauss constraints). Marc Henneaux For instance, for a 1-form in ﬁve dimensions, the dual form B is a Introduction 2-form and the duality symmetric action reads SO(2)-Electric- magnetic gravitational µ µ ¶ ¶ duality in D 4 Z 1 1 1 = S[A ,B ] dx0d4x ²ijkm H A˙ F B˙ H SO(2)-Electric- i jk jkm i ij km magnetic duality = 2 3! − 4 − for higher spins in D 4 with F dA, H dB and = = = Twisted self-duality µ ¶ 1 1 jkm 1 ij Conclusions H H H F Fij = 2 3! jkm + 2

17 / 20 Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

Twisted self-duality

Hamiltonian formalism for gauge systems They can be obtained by following the same procedure (go and manifestly duality Hamiltonian and introduce the second potential through solving symmetric actions the Gauss constraints). Marc Henneaux For instance, for a 1-form in ﬁve dimensions, the dual form B is a Introduction 2-form and the duality symmetric action reads SO(2)-Electric- magnetic gravitational µ µ ¶ ¶ duality in D 4 Z 1 1 1 = S[A ,B ] dx0d4x ²ijkm H A˙ F B˙ H SO(2)-Electric- i jk jkm i ij km magnetic duality = 2 3! − 4 − for higher spins in D 4 with F dA, H dB and = = = Twisted self-duality µ ¶ 1 1 jkm 1 ij Conclusions H H H F Fij = 2 3! jkm + 2

Chern-Simons couplings can be included.

17 / 20 Twisted self-duality

Hamiltonian formalism for gauge systems They can be obtained by following the same procedure (go and manifestly duality Hamiltonian and introduce the second potential through solving symmetric actions the Gauss constraints). Marc Henneaux For instance, for a 1-form in ﬁve dimensions, the dual form B is a Introduction 2-form and the duality symmetric action reads SO(2)-Electric- magnetic gravitational µ µ ¶ ¶ duality in D 4 Z 1 1 1 = S[A ,B ] dx0d4x ²ijkm H A˙ F B˙ H SO(2)-Electric- i jk jkm i ij km magnetic duality = 2 3! − 4 − for higher spins in D 4 with F dA, H dB and = = = Twisted self-duality µ ¶ 1 1 jkm 1 ij Conclusions H H H F Fij = 2 3! jkm + 2

Chern-Simons couplings can be included. Linearized gravity can also be treated (Bunster-Henneaux-Hörtner 2013).

17 / 20 If one wants to exhibit E10 or E11 as manifest symmetries of the action, one must be prepared to deal with actions that have less familiar properties. Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).

Conclusions

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

18 / 20 one must be prepared to deal with actions that have less familiar properties. Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).

Conclusions

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux If one wants to exhibit E10 or E11 as manifest symmetries of the Introduction action, SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

18 / 20 Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).

Conclusions

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux If one wants to exhibit E10 or E11 as manifest symmetries of the Introduction action, SO(2)-Electric- magnetic one must be prepared to deal with actions that have less familiar gravitational duality in D 4 = properties. SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

18 / 20 Duality invariance might be more fundamental (Bunster-Henneaux 2013).

Conclusions

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux If one wants to exhibit E10 or E11 as manifest symmetries of the Introduction action, SO(2)-Electric- magnetic one must be prepared to deal with actions that have less familiar gravitational duality in D 4 = properties. SO(2)-Electric- magnetic duality Manifestly duality invariant formulations do not exhibit manifest for higher spins in D 4 Poincaré invariance. = Twisted self-duality

Conclusions

18 / 20 Conclusions

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux If one wants to exhibit E10 or E11 as manifest symmetries of the Introduction action, SO(2)-Electric- magnetic one must be prepared to deal with actions that have less familiar gravitational duality in D 4 = properties. SO(2)-Electric- magnetic duality Manifestly duality invariant formulations do not exhibit manifest for higher spins in D 4 Poincaré invariance. = Twisted self-duality Duality invariance might be more fundamental

Conclusions (Bunster-Henneaux 2013).

18 / 20 Tension between manifest duality-invariance and manifest spacetime covariance. Does this tell us something ? Might duality invariance be more fundamental ? One may show that in the simple case of an Abelian vector ﬁeld (e.m.), duality invariance implies Poincaré invariance. The control of spacetime covariance is achieved through the Dirac-Schwinger commutation relations for the energy-momentum tensor components. The commutation relation

ij ¡ ¢ [H (x),H (x0)] δ H (x0) H (x) δ (x,x0) = i + i ,j is the only possibility for two conjugate transverse vectors~E,~B, and it implies Poincaré invariance. [Contains free Maxwell but also Born-Infeld.]

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

19 / 20 Does this tell us something ? Might duality invariance be more fundamental ? One may show that in the simple case of an Abelian vector ﬁeld (e.m.), duality invariance implies Poincaré invariance. The control of spacetime covariance is achieved through the Dirac-Schwinger commutation relations for the energy-momentum tensor components. The commutation relation

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

19 / 20 Might duality invariance be more fundamental ? One may show that in the simple case of an Abelian vector ﬁeld (e.m.), duality invariance implies Poincaré invariance. The control of spacetime covariance is achieved through the Dirac-Schwinger commutation relations for the energy-momentum tensor components. The commutation relation

Duality invariance and spacetime covariance

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

19 / 20 One may show that in the simple case of an Abelian vector ﬁeld (e.m.), duality invariance implies Poincaré invariance. The control of spacetime covariance is achieved through the Dirac-Schwinger commutation relations for the energy-momentum tensor components. The commutation relation

Duality invariance and spacetime covariance

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

19 / 20 The control of spacetime covariance is achieved through the Dirac-Schwinger commutation relations for the energy-momentum tensor components. The commutation relation

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Does this tell us something ? Marc Henneaux Might duality invariance be more fundamental ? Introduction One may show that in the simple case of an Abelian vector ﬁeld SO(2)-Electric- magnetic (e.m.), duality invariance implies Poincaré invariance. gravitational duality in D 4 = SO(2)-Electric- magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

19 / 20 The commutation relation

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Does this tell us something ? Marc Henneaux Might duality invariance be more fundamental ? Introduction One may show that in the simple case of an Abelian vector ﬁeld SO(2)-Electric- magnetic (e.m.), duality invariance implies Poincaré invariance. gravitational duality in D 4 = The control of spacetime covariance is achieved through the SO(2)-Electric- magnetic duality Dirac-Schwinger commutation relations for the for higher spins in D 4 energy-momentum tensor components. = Twisted self-duality

Conclusions

19 / 20 and it implies Poincaré invariance. [Contains free Maxwell but also Born-Infeld.]

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Does this tell us something ? Marc Henneaux Might duality invariance be more fundamental ? Introduction One may show that in the simple case of an Abelian vector ﬁeld SO(2)-Electric- magnetic (e.m.), duality invariance implies Poincaré invariance. gravitational duality in D 4 = The control of spacetime covariance is achieved through the SO(2)-Electric- magnetic duality Dirac-Schwinger commutation relations for the for higher spins in D 4 energy-momentum tensor components. = Twisted self-duality The commutation relation

Conclusions ij ¡ ¢ [H (x),H (x0)] δ H (x0) H (x) δ (x,x0) = i + i ,j is the only possibility for two conjugate transverse vectors~E,~B,

19 / 20 [Contains free Maxwell but also Born-Infeld.]

Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Does this tell us something ? Marc Henneaux Might duality invariance be more fundamental ? Introduction One may show that in the simple case of an Abelian vector ﬁeld SO(2)-Electric- magnetic (e.m.), duality invariance implies Poincaré invariance. gravitational duality in D 4 = The control of spacetime covariance is achieved through the SO(2)-Electric- magnetic duality Dirac-Schwinger commutation relations for the for higher spins in D 4 energy-momentum tensor components. = Twisted self-duality The commutation relation

Conclusions ij ¡ ¢ [H (x),H (x0)] δ H (x0) H (x) δ (x,x0) = i + i ,j is the only possibility for two conjugate transverse vectors~E,~B, and it implies Poincaré invariance.

19 / 20 Duality invariance and spacetime covariance

Hamiltonian formalism for gauge systems Tension between manifest duality-invariance and manifest and manifestly duality spacetime covariance. symmetric actions Does this tell us something ? Marc Henneaux Might duality invariance be more fundamental ? Introduction One may show that in the simple case of an Abelian vector ﬁeld SO(2)-Electric- magnetic (e.m.), duality invariance implies Poincaré invariance. gravitational duality in D 4 = The control of spacetime covariance is achieved through the SO(2)-Electric- magnetic duality Dirac-Schwinger commutation relations for the for higher spins in D 4 energy-momentum tensor components. = Twisted self-duality The commutation relation

Conclusions ij ¡ ¢ [H (x),H (x0)] δ H (x0) H (x) δ (x,x0) = i + i ,j is the only possibility for two conjugate transverse vectors~E,~B, and it implies Poincaré invariance. [Contains free Maxwell but also Born-Infeld.]

19 / 20 Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux

Introduction

SO(2)-Electric- magnetic gravitational duality in D 4 = SO(2)-Electric- HAPPY BIRTHDAY, QUIM ! magnetic duality for higher spins in D 4 = Twisted self-duality

Conclusions

20 / 20