Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational Gravitational electric-magnetic duality duality in D 4 (linearized = gravity)
Twisted self-duality for Marc Henneaux p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance November 2013 and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Electric-magnetic duality is a fascinating symmetry. Originally considered in the context of electromagnetism, it also plays a key role in extended supergravity models, where the duality group (acting on the vector fields and the scalars) is enlarged to U(n) or Sp(2n,R). In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Originally considered in the context of electromagnetism, it also plays a key role in extended supergravity models, where the duality group (acting on the vector fields and the scalars) is enlarged to U(n) or Sp(2n,R). In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 = Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized = enlarged to U(n) or Sp(2n,R). gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 = Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized = enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 = Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized = enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms
Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 = Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized = enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms
Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravitational electric-magnetic duality (acting on the graviton) is Gravity
Duality invariance also very intriguing. It rotates for instance the Schwarzchild mass and spacetime covariance into the Taub-NUT parameter N.
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Introduction
Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry.
Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 = Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized = enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms
Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravitational electric-magnetic duality (acting on the graviton) is Gravity
Duality invariance also very intriguing. It rotates for instance the Schwarzchild mass and spacetime covariance into the Taub-NUT parameter N. Conclusions Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries".
Marc Henneaux Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10
8
−1 0 1 2 3 4 5 6 7
or E11 11
10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Introduction
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 8
−1 0 1 2 3 4 5 6 7
or E11 11
10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Introduction
Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality or E11 11
10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Introduction
Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity) −1 0 1 2 3 4 5 6 7
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 11
10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Introduction
Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity) −1 0 1 2 3 4 5 6 7
Twisted self-duality for or E11 p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Introduction
Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity) −1 0 1 2 3 4 5 6 7
Twisted self-duality for or E11 p-forms 11 Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime 10 9 8 7 6 5 4 3 2 1 covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Introduction
Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity) −1 0 1 2 3 4 5 6 7
Twisted self-duality for or E11 p-forms 11 Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime 10 9 8 7 6 5 4 3 2 1 covariance
Conclusions which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it.
Marc Henneaux Gravitational electric-magnetic duality One crucial feature of these algebras is that they treat democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, 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. Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Whenever a (dynamical) p-form gauge field appears, 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. Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals.
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Similarly, the graviton and its dual, described by a field with Young symmetry
D 3 boxes − simultaneously appear. Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, its dual Introduction
Electromagnetism D p 2-form gauge field also appears. in D 4 − − = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality D 3 boxes − simultaneously appear. Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, its dual Introduction
Electromagnetism D p 2-form gauge field also appears. in D 4 − − = Similarly, the graviton and its dual, described by a field with Gravitational duality in D 4 Young symmetry (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality simultaneously appear. Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, its dual Introduction
Electromagnetism D p 2-form gauge field also appears. in D 4 − − = Similarly, the graviton and its dual, described by a field with Gravitational duality in D 4 Young symmetry (linearized = gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) D 3 boxes Gravity − Duality invariance and spacetime covariance Conclusions
Marc Henneaux Gravitational electric-magnetic duality Understanding gravitational duality is thus important in this context.
Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, its dual Introduction
Electromagnetism D p 2-form gauge field also appears. in D 4 − − = Similarly, the graviton and its dual, described by a field with Gravitational duality in D 4 Young symmetry (linearized = gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) D 3 boxes Gravity − Duality invariance and spacetime covariance Conclusions simultaneously appear.
Marc Henneaux Gravitational electric-magnetic duality Introduction
Gravitational electric-magnetic One crucial feature of these algebras is that they treat duality
Marc Henneaux democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears, its dual Introduction
Electromagnetism D p 2-form gauge field also appears. in D 4 − − = Similarly, the graviton and its dual, described by a field with Gravitational duality in D 4 Young symmetry (linearized = gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) D 3 boxes Gravity − Duality invariance and spacetime covariance Conclusions simultaneously appear. Understanding gravitational duality is thus important in this context.
Marc Henneaux Gravitational electric-magnetic duality discuss first electric-magnetic duality in its original D 4 = electromagnetic context...... and show in particular that contrary to widespread (and incorrect) belief, duality is a symmetry of the Maxwell action and not just of the equations of motion ; explain next gravitational duality at the linearized level again in D 4 ; = show then that in D 4, what generalizes duality invariance is > “twisted self-duality", which puts each field and its dual on an equal footing ; finally conclude and mention some open questions.
Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality ... and show in particular that contrary to widespread (and incorrect) belief, duality is a symmetry of the Maxwell action and not just of the equations of motion ; explain next gravitational duality at the linearized level again in D 4 ; = show then that in D 4, what generalizes duality invariance is > “twisted self-duality", which puts each field and its dual on an equal footing ; finally conclude and mention some open questions.
Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction Electromagnetism discuss first electric-magnetic duality in its original D 4 in D 4 = = electromagnetic context... Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality explain next gravitational duality at the linearized level again in D 4 ; = show then that in D 4, what generalizes duality invariance is > “twisted self-duality", which puts each field and its dual on an equal footing ; finally conclude and mention some open questions.
Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction Electromagnetism discuss first electric-magnetic duality in its original D 4 in D 4 = = electromagnetic context... Gravitational duality in D 4 (linearized = ... and show in particular that contrary to widespread (and gravity) incorrect) belief, duality is a symmetry of the Maxwell action Twisted self-duality for and not just of the equations of motion ; p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality show then that in D 4, what generalizes duality invariance is > “twisted self-duality", which puts each field and its dual on an equal footing ; finally conclude and mention some open questions.
Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction Electromagnetism discuss first electric-magnetic duality in its original D 4 in D 4 = = electromagnetic context... Gravitational duality in D 4 (linearized = ... and show in particular that contrary to widespread (and gravity) incorrect) belief, duality is a symmetry of the Maxwell action Twisted self-duality for and not just of the equations of motion ; p-forms
Twisted explain next gravitational duality at the linearized level again in self-duality for (linearized) D 4 ; Gravity =
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality finally conclude and mention some open questions.
Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction Electromagnetism discuss first electric-magnetic duality in its original D 4 in D 4 = = electromagnetic context... Gravitational duality in D 4 (linearized = ... and show in particular that contrary to widespread (and gravity) incorrect) belief, duality is a symmetry of the Maxwell action Twisted self-duality for and not just of the equations of motion ; p-forms
Twisted explain next gravitational duality at the linearized level again in self-duality for (linearized) D 4 ; Gravity = show then that in D 4, what generalizes duality invariance is Duality invariance > and spacetime “twisted self-duality", which puts each field and its dual on an covariance
Conclusions equal footing ;
Marc Henneaux Gravitational electric-magnetic duality Introduction
Gravitational electric-magnetic duality The purpose of this talk is to : Marc Henneaux
Introduction Electromagnetism discuss first electric-magnetic duality in its original D 4 in D 4 = = electromagnetic context... Gravitational duality in D 4 (linearized = ... and show in particular that contrary to widespread (and gravity) incorrect) belief, duality is a symmetry of the Maxwell action Twisted self-duality for and not just of the equations of motion ; p-forms
Twisted explain next gravitational duality at the linearized level again in self-duality for (linearized) D 4 ; Gravity = show then that in D 4, what generalizes duality invariance is Duality invariance > and spacetime “twisted self-duality", which puts each field and its dual on an covariance
Conclusions equal footing ; finally conclude and mention some open questions.
Marc Henneaux Gravitational electric-magnetic duality The duality transformations are usually written in terms of the field strengths
µν µν µν F cosαF sinα ∗F → − µν µν µν ∗F sinαF cosα ∗F , → +
or in (3 1)- fashion, +
E cosαE sinαB → + B sinαE cosαB. → − +
EM duality as an off-shell symmetry of the Maxwell theory in D 4 =
Gravitational electric-magnetic duality We start with the Maxwell theory in 4 dimensions. Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality µν µν µν F cosαF sinα ∗F → − µν µν µν ∗F sinαF cosα ∗F , → +
or in (3 1)- fashion, +
E cosαE sinαB → + B sinαE cosαB. → − +
EM duality as an off-shell symmetry of the Maxwell theory in D 4 =
Gravitational electric-magnetic duality We start with the Maxwell theory in 4 dimensions. Marc Henneaux The duality transformations are usually written in terms of the Introduction
Electromagnetism field strengths in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality or in (3 1)- fashion, +
E cosαE sinαB → + B sinαE cosαB. → − +
EM duality as an off-shell symmetry of the Maxwell theory in D 4 =
Gravitational electric-magnetic duality We start with the Maxwell theory in 4 dimensions. Marc Henneaux The duality transformations are usually written in terms of the Introduction
Electromagnetism field strengths in D 4 = Gravitational µν µν µν duality in D 4 F cosαF sinα ∗F (linearized = → − gravity) Fµν sinαFµν cosα Fµν, Twisted ∗ ∗ self-duality for → + p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality E cosαE sinαB → + B sinαE cosαB. → − +
EM duality as an off-shell symmetry of the Maxwell theory in D 4 =
Gravitational electric-magnetic duality We start with the Maxwell theory in 4 dimensions. Marc Henneaux The duality transformations are usually written in terms of the Introduction
Electromagnetism field strengths in D 4 = Gravitational µν µν µν duality in D 4 F cosαF sinα ∗F (linearized = → − gravity) Fµν sinαFµν cosα Fµν, Twisted ∗ ∗ self-duality for → + p-forms
Twisted self-duality for or in (3 1)- fashion, (linearized) + Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality EM duality as an off-shell symmetry of the Maxwell theory in D 4 =
Gravitational electric-magnetic duality We start with the Maxwell theory in 4 dimensions. Marc Henneaux The duality transformations are usually written in terms of the Introduction
Electromagnetism field strengths in D 4 = Gravitational µν µν µν duality in D 4 F cosαF sinα ∗F (linearized = → − gravity) Fµν sinαFµν cosα Fµν, Twisted ∗ ∗ self-duality for → + p-forms
Twisted self-duality for or in (3 1)- fashion, (linearized) + Gravity
Duality invariance and spacetime E cosαE sinαB covariance → +
Conclusions B sinαE cosαB. → − +
Marc Henneaux Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell equations dF 0, d∗F 0, or = =
E 0, B 0, ∇ · = ∇ · = ∂E ∂B B 0, E 0, ∂t − ∇ × = ∂t + ∇ × =
invariant. We claim that these transformations also leave the Maxwell action
Z Z 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) = −4 = 2 −
invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality E 0, B 0, ∇ · = ∇ · = ∂E ∂B B 0, E 0, ∂t − ∇ × = ∂t + ∇ × =
invariant. We claim that these transformations also leave the Maxwell action
Z Z 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) = −4 = 2 −
invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality invariant. We claim that these transformations also leave the Maxwell action
Z Z 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) = −4 = 2 −
invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 E 0, B 0, = ∇ · = ∇ · = Gravitational duality in D 4 ∂E ∂B (linearized = B 0, E 0, gravity) ∂t − ∇ × = ∂t + ∇ × =
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality We claim that these transformations also leave the Maxwell action
Z Z 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) = −4 = 2 −
invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 E 0, B 0, = ∇ · = ∇ · = Gravitational duality in D 4 ∂E ∂B (linearized = B 0, E 0, gravity) ∂t − ∇ × = ∂t + ∇ × =
Twisted self-duality for p-forms invariant. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Z Z 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) = −4 = 2 −
invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 E 0, B 0, = ∇ · = ∇ · = Gravitational duality in D 4 ∂E ∂B (linearized = B 0, E 0, gravity) ∂t − ∇ × = ∂t + ∇ × =
Twisted self-duality for p-forms invariant. Twisted self-duality for We claim that these transformations also leave the Maxwell (linearized) Gravity action Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality invariant.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 E 0, B 0, = ∇ · = ∇ · = Gravitational duality in D 4 ∂E ∂B (linearized = B 0, E 0, gravity) ∂t − ∇ × = ∂t + ∇ × =
Twisted self-duality for p-forms invariant. Twisted self-duality for We claim that these transformations also leave the Maxwell (linearized) Gravity action Duality invariance and spacetime Z Z covariance 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) Conclusions = −4 = 2 −
Marc Henneaux Gravitational electric-magnetic duality EM duality as an off-shell symmetry
Gravitational electric-magnetic duality These transformations are well known to leave the Maxwell Marc Henneaux equations dF 0, d∗F 0, or = = Introduction
Electromagnetism in D 4 E 0, B 0, = ∇ · = ∇ · = Gravitational duality in D 4 ∂E ∂B (linearized = B 0, E 0, gravity) ∂t − ∇ × = ∂t + ∇ × =
Twisted self-duality for p-forms invariant. Twisted self-duality for We claim that these transformations also leave the Maxwell (linearized) Gravity action Duality invariance and spacetime Z Z covariance 1 4 µν 1 4 2 2 S d xFµν ∗F d x(E B ) Conclusions = −4 = 2 −
invariant.
Marc Henneaux Gravitational electric-magnetic duality As a by-product, we shall also clarify why the hyperbolic rotations in the (E,B)-plane
E coshαE sinhαB → + B sinhαE coshαB, → +
which seemingly leave the Maxwell action invariant, are not symmetries of the theory and in particular are not symmetries of the equations of motion. There is no contradiction with Noether’s theorem.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality E coshαE sinhαB → + B sinhαE coshαB, → +
which seemingly leave the Maxwell action invariant, are not symmetries of the theory and in particular are not symmetries of the equations of motion. There is no contradiction with Noether’s theorem.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction As a by-product, we shall also clarify why the hyperbolic rotations
Electromagnetism in the (E,B)-plane in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality which seemingly leave the Maxwell action invariant, are not symmetries of the theory and in particular are not symmetries of the equations of motion. There is no contradiction with Noether’s theorem.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction As a by-product, we shall also clarify why the hyperbolic rotations
Electromagnetism in the (E,B)-plane in D 4 = Gravitational duality in D 4 (linearized = E coshαE sinhαB gravity) → +
Twisted B sinhαE coshαB, self-duality for → + p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality are not symmetries of the theory and in particular are not symmetries of the equations of motion. There is no contradiction with Noether’s theorem.
EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction As a by-product, we shall also clarify why the hyperbolic rotations
Electromagnetism in the (E,B)-plane in D 4 = Gravitational duality in D 4 (linearized = E coshαE sinhαB gravity) → +
Twisted B sinhαE coshαB, self-duality for → + p-forms
Twisted self-duality for (linearized) which seemingly leave the Maxwell action invariant, Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality EM duality as an off-shell symmetry
Gravitational electric-magnetic duality
Marc Henneaux
Introduction As a by-product, we shall also clarify why the hyperbolic rotations
Electromagnetism in the (E,B)-plane in D 4 = Gravitational duality in D 4 (linearized = E coshαE sinhαB gravity) → +
Twisted B sinhαE coshαB, self-duality for → + p-forms
Twisted self-duality for (linearized) which seemingly leave the Maxwell action invariant, Gravity are not symmetries of the theory and in particular are not Duality invariance and spacetime symmetries of the equations of motion. There is no contradiction covariance with Noether’s theorem. Conclusions
Marc Henneaux Gravitational electric-magnetic duality Since the dynamical variables that are varied in the action principle are the components Aµ of the vector potential, one needs to express the duality transformations in terms of Aµ, with F ∂ A ∂ A µν = µ ν − ν µ (so that B A). = ∇ × Furthermore, one must know these transformations off-shell since one must go off-shell to check invariance of the action. But one encounters the following problem !
Theorem : There is no variation of Aµ that yields the above duality transformations of the field strengths.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality with F ∂ A ∂ A µν = µ ν − ν µ (so that B A). = ∇ × Furthermore, one must know these transformations off-shell since one must go off-shell to check invariance of the action. But one encounters the following problem !
Theorem : There is no variation of Aµ that yields the above duality transformations of the field strengths.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux Since the dynamical variables that are varied in the action Introduction principle are the components Aµ of the vector potential, one Electromagnetism in D 4 = needs to express the duality transformations in terms of Aµ, Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Furthermore, one must know these transformations off-shell since one must go off-shell to check invariance of the action. But one encounters the following problem !
Theorem : There is no variation of Aµ that yields the above duality transformations of the field strengths.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux Since the dynamical variables that are varied in the action Introduction principle are the components Aµ of the vector potential, one Electromagnetism in D 4 = needs to express the duality transformations in terms of Aµ, Gravitational duality in D 4 with (linearized = gravity) F ∂ A ∂ A µν = µ ν − ν µ Twisted self-duality for (so that B A). p-forms = ∇ × Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality But one encounters the following problem !
Theorem : There is no variation of Aµ that yields the above duality transformations of the field strengths.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux Since the dynamical variables that are varied in the action Introduction principle are the components Aµ of the vector potential, one Electromagnetism in D 4 = needs to express the duality transformations in terms of Aµ, Gravitational duality in D 4 with (linearized = gravity) F ∂ A ∂ A µν = µ ν − ν µ Twisted self-duality for (so that B A). p-forms = ∇ × Twisted Furthermore, one must know these transformations off-shell self-duality for (linearized) since one must go off-shell to check invariance of the action. Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Theorem : There is no variation of Aµ that yields the above duality transformations of the field strengths.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux Since the dynamical variables that are varied in the action Introduction principle are the components Aµ of the vector potential, one Electromagnetism in D 4 = needs to express the duality transformations in terms of Aµ, Gravitational duality in D 4 with (linearized = gravity) F ∂ A ∂ A µν = µ ν − ν µ Twisted self-duality for (so that B A). p-forms = ∇ × Twisted Furthermore, one must know these transformations off-shell self-duality for (linearized) since one must go off-shell to check invariance of the action. Gravity
Duality invariance But one encounters the following problem ! and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux Since the dynamical variables that are varied in the action Introduction principle are the components Aµ of the vector potential, one Electromagnetism in D 4 = needs to express the duality transformations in terms of Aµ, Gravitational duality in D 4 with (linearized = gravity) F ∂ A ∂ A µν = µ ν − ν µ Twisted self-duality for (so that B A). p-forms = ∇ × Twisted Furthermore, one must know these transformations off-shell self-duality for (linearized) since one must go off-shell to check invariance of the action. Gravity
Duality invariance But one encounters the following problem ! and spacetime covariance Theorem : There is no variation of Aµ that yields the above Conclusions duality transformations of the field strengths.
Marc Henneaux Gravitational electric-magnetic duality The proof is elementary.
Once A is introduced, dF 0 is an identity. But dF0 0 does not µ = = hold off-shell (unless α is a multiple of π but then F and its dual are not mixed), where F0 is the new field strength cosαF sinα ∗F − after duality rotation. Hence there is no A0 such that F0 dA0. = It follows from this theorem that it is meaningless to ask whether the Maxwell action S[Aµ] is invariant under the above duality transformations of the field strengths since there is no variation of Aµ that yields these variations. In the same way, it is meaningless to ask whether the Maxwell action is invariant under hyperbolic rotations (and conclude that it is, yielding an apparent contradiction with the non-invariance of the equations of motion), because the same argument indicates that there is no variation of the vector potential that gives the hyperbolic rotations off-shell for the field strength.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Once A is introduced, dF 0 is an identity. But dF0 0 does not µ = = hold off-shell (unless α is a multiple of π but then F and its dual are not mixed), where F0 is the new field strength cosαF sinα ∗F − after duality rotation. Hence there is no A0 such that F0 dA0. = It follows from this theorem that it is meaningless to ask whether the Maxwell action S[Aµ] is invariant under the above duality transformations of the field strengths since there is no variation of Aµ that yields these variations. In the same way, it is meaningless to ask whether the Maxwell action is invariant under hyperbolic rotations (and conclude that it is, yielding an apparent contradiction with the non-invariance of the equations of motion), because the same argument indicates that there is no variation of the vector potential that gives the hyperbolic rotations off-shell for the field strength.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality The proof is elementary. Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality It follows from this theorem that it is meaningless to ask whether the Maxwell action S[Aµ] is invariant under the above duality transformations of the field strengths since there is no variation of Aµ that yields these variations. In the same way, it is meaningless to ask whether the Maxwell action is invariant under hyperbolic rotations (and conclude that it is, yielding an apparent contradiction with the non-invariance of the equations of motion), because the same argument indicates that there is no variation of the vector potential that gives the hyperbolic rotations off-shell for the field strength.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality The proof is elementary. Marc Henneaux Once Aµ is introduced, dF 0 is an identity. But dF0 0 does not Introduction = = hold off-shell (unless α is a multiple of π but then F and its dual Electromagnetism in D 4 are not mixed), where F0 is the new field strength cosαF sinα ∗F = − Gravitational after duality rotation. Hence there is no A such that F dA . duality in D 4 0 0 0 (linearized = = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality In the same way, it is meaningless to ask whether the Maxwell action is invariant under hyperbolic rotations (and conclude that it is, yielding an apparent contradiction with the non-invariance of the equations of motion), because the same argument indicates that there is no variation of the vector potential that gives the hyperbolic rotations off-shell for the field strength.
Duality transformations of the potential - The problem
Gravitational electric-magnetic duality The proof is elementary. Marc Henneaux Once Aµ is introduced, dF 0 is an identity. But dF0 0 does not Introduction = = hold off-shell (unless α is a multiple of π but then F and its dual Electromagnetism in D 4 are not mixed), where F0 is the new field strength cosαF sinα ∗F = − Gravitational after duality rotation. Hence there is no A such that F dA . duality in D 4 0 0 0 (linearized = = gravity) It follows from this theorem that it is meaningless to ask whether Twisted the Maxwell action S[Aµ] is invariant under the above duality self-duality for p-forms transformations of the field strengths since there is no Twisted variation of Aµ that yields these variations. self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality transformations of the potential - The problem
Gravitational electric-magnetic duality The proof is elementary. Marc Henneaux Once Aµ is introduced, dF 0 is an identity. But dF0 0 does not Introduction = = hold off-shell (unless α is a multiple of π but then F and its dual Electromagnetism in D 4 are not mixed), where F0 is the new field strength cosαF sinα ∗F = − Gravitational after duality rotation. Hence there is no A such that F dA . duality in D 4 0 0 0 (linearized = = gravity) It follows from this theorem that it is meaningless to ask whether Twisted the Maxwell action S[Aµ] is invariant under the above duality self-duality for p-forms transformations of the field strengths since there is no Twisted variation of Aµ that yields these variations. self-duality for (linearized) Gravity In the same way, it is meaningless to ask whether the Maxwell
Duality invariance action is invariant under hyperbolic rotations (and conclude that and spacetime covariance it is, yielding an apparent contradiction with the non-invariance Conclusions of the equations of motion), because the same argument indicates that there is no variation of the vector potential that gives the hyperbolic rotations off-shell for the field strength.
Marc Henneaux Gravitational electric-magnetic duality Although there is no variation of the vector potential that yields the standard duality rotations of the field strengths off-shell, one can find transformations of Aµ that reproduce them on-shell. As we just argued, this is the best one can hope for.
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality As we just argued, this is the best one can hope for.
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 Although there is no variation of the vector potential that yields (linearized = gravity) the standard duality rotations of the field strengths off-shell, one Twisted can find transformations of Aµ that reproduce them on-shell. self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 Although there is no variation of the vector potential that yields (linearized = gravity) the standard duality rotations of the field strengths off-shell, one Twisted can find transformations of Aµ that reproduce them on-shell. self-duality for p-forms As we just argued, this is the best one can hope for. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Going from the transformations of the field strength to the transformations of the vector potential requires integrations and introduces non-local terms. We shall choose the duality transformations of the vector potential such that these non-local terms are non-local only in 1 space, where the inverse Laplacian − can be given a meaning. 1 4 Terms non local in time (and − ) are much more tricky. We also ä use the gauge ambiguity in the definition of δAµ in such a way that δA 0. 0 = The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by
i 1 ³ ijk ´ δA 0, δA ² − ² ∂ F . 0 = = − 4 j 0k i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = − 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality We shall choose the duality transformations of the vector potential such that these non-local terms are non-local only in 1 space, where the inverse Laplacian − can be given a meaning. 1 4 Terms non local in time (and − ) are much more tricky. We also ä use the gauge ambiguity in the definition of δAµ in such a way that δA 0. 0 = The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by
i 1 ³ ijk ´ δA 0, δA ² − ² ∂ F . 0 = = − 4 j 0k i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = − 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic Going from the transformations of the field strength to the duality
Marc Henneaux transformations of the vector potential requires integrations and introduces non-local terms. Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by
i 1 ³ ijk ´ δA 0, δA ² − ² ∂ F . 0 = = − 4 j 0k i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = − 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic Going from the transformations of the field strength to the duality
Marc Henneaux transformations of the vector potential requires integrations and introduces non-local terms. Introduction
Electromagnetism We shall choose the duality transformations of the vector in D 4 = potential such that these non-local terms are non-local only in Gravitational 1 duality in D 4 space, where the inverse Laplacian − can be given a meaning. (linearized = 1 4 gravity) Terms non local in time (and − ) are much more tricky. We also ä Twisted use the gauge ambiguity in the definition of δAµ in such a way self-duality for p-forms that δA 0. 0 = Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = − 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic Going from the transformations of the field strength to the duality
Marc Henneaux transformations of the vector potential requires integrations and introduces non-local terms. Introduction
Electromagnetism We shall choose the duality transformations of the vector in D 4 = potential such that these non-local terms are non-local only in Gravitational 1 duality in D 4 space, where the inverse Laplacian − can be given a meaning. (linearized = 1 4 gravity) Terms non local in time (and − ) are much more tricky. We also ä Twisted use the gauge ambiguity in the definition of δAµ in such a way self-duality for p-forms that δA 0. 0 = Twisted self-duality for The duality transformation of the vector potential is then given (linearized) Gravity (up to a residual gauge symmetry) by
Duality invariance i 1 ³ ijk ´ and spacetime δA0 0, δA ² − ² ∂jF0k . covariance = = − 4 Conclusions
Marc Henneaux Gravitational electric-magnetic duality 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ
Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic Going from the transformations of the field strength to the duality
Marc Henneaux transformations of the vector potential requires integrations and introduces non-local terms. Introduction
Electromagnetism We shall choose the duality transformations of the vector in D 4 = potential such that these non-local terms are non-local only in Gravitational 1 duality in D 4 space, where the inverse Laplacian − can be given a meaning. (linearized = 1 4 gravity) Terms non local in time (and − ) are much more tricky. We also ä Twisted use the gauge ambiguity in the definition of δAµ in such a way self-duality for p-forms that δA 0. 0 = Twisted self-duality for The duality transformation of the vector potential is then given (linearized) Gravity (up to a residual gauge symmetry) by
Duality invariance i 1 ³ ijk ´ and spacetime δA0 0, δA ² − ² ∂jF0k . covariance = = − 4 Conclusions i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = −
Marc Henneaux Gravitational electric-magnetic duality Explicit form of the duality transformation of the vector potential under em duality
Gravitational electric-magnetic Going from the transformations of the field strength to the duality
Marc Henneaux transformations of the vector potential requires integrations and introduces non-local terms. Introduction
Electromagnetism We shall choose the duality transformations of the vector in D 4 = potential such that these non-local terms are non-local only in Gravitational 1 duality in D 4 space, where the inverse Laplacian − can be given a meaning. (linearized = 1 4 gravity) Terms non local in time (and − ) are much more tricky. We also ä Twisted use the gauge ambiguity in the definition of δAµ in such a way self-duality for p-forms that δA 0. 0 = Twisted self-duality for The duality transformation of the vector potential is then given (linearized) Gravity (up to a residual gauge symmetry) by
Duality invariance i 1 ³ ijk ´ and spacetime δA0 0, δA ² − ² ∂jF0k . covariance = = − 4 Conclusions i i 1 i j i 1 i µ It implies δB ²E ² − (∂ ∂ F ) ²E ² − (∂ ∂ F ), i.e., = − − 4 0j = − − 4 0µ δBi ²Ei on-shell. = − 1 j µk Similarly, δE δA˙ ²B ² − (² ∂ ∂ F ). i = − i = i − 4 ijk µ Marc Henneaux Gravitational electric-magnetic duality The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Indeed, one finds µ Z ¶ µ Z ¶ 1 3 2 d 1 3 ijk 1 ¡ ¢ δ d xE ² d xEi² − ∂jE 2 = dt − 2 4 k
and µ Z ¶ µ Z ¶ 1 3 2 d 1 3 ijk 1 ¡ ¢ δ d xB ² d xBi² − ∂jB 2 = dt − 2 4 k
so that δS δR dtL 0 (with L 1 R d3x¡E2 B2¢). = = = 2 − It is therefore a genuine Noether symmetry (with Noether charge etc).
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Indeed, one finds µ Z ¶ µ Z ¶ 1 3 2 d 1 3 ijk 1 ¡ ¢ δ d xE ² d xEi² − ∂jE 2 = dt − 2 4 k
and µ Z ¶ µ Z ¶ 1 3 2 d 1 3 ijk 1 ¡ ¢ δ d xB ² d xBi² − ∂jB 2 = dt − 2 4 k
so that δS δR dtL 0 (with L 1 R d3x¡E2 B2¢). = = = 2 − It is therefore a genuine Noether symmetry (with Noether charge etc).
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality and µ Z ¶ µ Z ¶ 1 3 2 d 1 3 ijk 1 ¡ ¢ δ d xB ² d xBi² − ∂jB 2 = dt − 2 4 k
so that δS δR dtL 0 (with L 1 R d3x¡E2 B2¢). = = = 2 − It is therefore a genuine Noether symmetry (with Noether charge etc).
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Introduction
Electromagnetism Indeed, one finds in D 4 = Gravitational µ 1 Z ¶ d µ 1 Z ¶ duality in D 4 3 2 3 ijk 1 ¡ ¢ (linearized = δ d xE ² d xEi² − ∂jEk gravity) 2 = dt − 2 4
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality so that δS δR dtL 0 (with L 1 R d3x¡E2 B2¢). = = = 2 − It is therefore a genuine Noether symmetry (with Noether charge etc).
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Introduction
Electromagnetism Indeed, one finds in D 4 = Gravitational µ 1 Z ¶ d µ 1 Z ¶ duality in D 4 3 2 3 ijk 1 ¡ ¢ (linearized = δ d xE ² d xEi² − ∂jEk gravity) 2 = dt − 2 4
Twisted self-duality for p-forms and Twisted µ 1 Z ¶ d µ 1 Z ¶ self-duality for 3 2 3 ijk 1 ¡ ¢ (linearized) δ d xB ² d xBi² − ∂jBk Gravity 2 = dt − 2 4
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality It is therefore a genuine Noether symmetry (with Noether charge etc).
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Introduction
Electromagnetism Indeed, one finds in D 4 = Gravitational µ 1 Z ¶ d µ 1 Z ¶ duality in D 4 3 2 3 ijk 1 ¡ ¢ (linearized = δ d xE ² d xEi² − ∂jEk gravity) 2 = dt − 2 4
Twisted self-duality for p-forms and Twisted µ 1 Z ¶ d µ 1 Z ¶ self-duality for 3 2 3 ijk 1 ¡ ¢ (linearized) δ d xB ² d xBi² − ∂jBk Gravity 2 = dt − 2 4
Duality invariance and spacetime covariance R 1 R 3 ¡ 2 2¢ so that δS δ dtL 0 (with L 2 d x E B ). Conclusions = = = −
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. Introduction
Electromagnetism Indeed, one finds in D 4 = Gravitational µ 1 Z ¶ d µ 1 Z ¶ duality in D 4 3 2 3 ijk 1 ¡ ¢ (linearized = δ d xE ² d xEi² − ∂jEk gravity) 2 = dt − 2 4
Twisted self-duality for p-forms and Twisted µ 1 Z ¶ d µ 1 Z ¶ self-duality for 3 2 3 ijk 1 ¡ ¢ (linearized) δ d xB ² d xBi² − ∂jBk Gravity 2 = dt − 2 4
Duality invariance and spacetime R 1 R 3 ¡ 2 2¢ covariance so that δS δ dtL 0 (with L d x E B ). = = = 2 − Conclusions It is therefore a genuine Noether symmetry (with Noether charge etc).
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action is manifest if one goes to the first-order form and introduces a second vector-potential by solving Gauss’ constraint E 0. ∇ · = If, besides the standard “magnetic" vector potential defined through, ~B ~B ~ ~A , ≡ 1 = ∇ × 1 one introduces an additional vector potential~A2 through,
~E ~B ~ ~A , ≡ 2 = ∇ × 2 one may rewrite the standard Maxwell action in terms of the two potentials Aa as
1 Z ³ ´ S dx0d3x ² ~Ba ~A˙ b δ ~Ba ~Bb . = 2 ab · − ab ·
Here, ² is given by ² ² , ² 1. ab ab = − ba 12 = +
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality If, besides the standard “magnetic" vector potential defined through, ~B ~B ~ ~A , ≡ 1 = ∇ × 1 one introduces an additional vector potential~A2 through,
~E ~B ~ ~A , ≡ 2 = ∇ × 2 one may rewrite the standard Maxwell action in terms of the two potentials Aa as
1 Z ³ ´ S dx0d3x ² ~Ba ~A˙ b δ ~Ba ~Bb . = 2 ab · − ab ·
Here, ² is given by ² ² , ² 1. ab ab = − ba 12 = +
Duality invariance of the action
Gravitational electric-magnetic duality Duality invariance of the action is manifest if one goes to the Marc Henneaux first-order form and introduces a second vector-potential by solving Gauss’ constraint E 0. Introduction ∇ · = Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action
Gravitational electric-magnetic duality Duality invariance of the action is manifest if one goes to the Marc Henneaux first-order form and introduces a second vector-potential by solving Gauss’ constraint E 0. Introduction ∇ · = Electromagnetism If, besides the standard “magnetic" vector potential defined in D 4 = Gravitational through, duality in D 4 ~ ~ ~ ~ (linearized = B B1 A1, gravity) ≡ = ∇ × ~ Twisted one introduces an additional vector potential A2 through, self-duality for p-forms ~ ~ ~ ~ Twisted E B2 A2, self-duality for ≡ = ∇ × (linearized) Gravity one may rewrite the standard Maxwell action in terms of the two Duality invariance a and spacetime potentials A as covariance Z Conclusions 1 ³ ´ S dx0d3x ² ~Ba ~A˙ b δ ~Ba ~Bb . = 2 ab · − ab ·
Here, ² is given by ² ² , ² 1. ab ab = − ba 12 = +
Marc Henneaux Gravitational electric-magnetic duality The action is invariant under rotations in the (1,2) plane of the vector potentials (“electric-magnetic duality rotations") because ²ab and δab are invariant tensors. The action is also invariant under the gauge transformations,
~Aa ~Aa ~ Λa. −→ + ∇
To conclude : the “proof" using the standard form of the em duality transformations that the second order Maxwell action S[A ] 1 R d4xF Fµν is not invariant under duality µ = − 4 µν transformations is simply incorrect because it is based on a form of the duality transformations that is inconsistent with the existence of the dynamical variable Aµ. A consistent set of transformations leaves the action invariant. Deser-Teitelboim 1976, Deser-Gomberoff-Henneaux-Teitelboim 1997
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The action is also invariant under the gauge transformations,
~Aa ~Aa ~ Λa. −→ + ∇
To conclude : the “proof" using the standard form of the em duality transformations that the second order Maxwell action S[A ] 1 R d4xF Fµν is not invariant under duality µ = − 4 µν transformations is simply incorrect because it is based on a form of the duality transformations that is inconsistent with the existence of the dynamical variable Aµ. A consistent set of transformations leaves the action invariant. Deser-Teitelboim 1976, Deser-Gomberoff-Henneaux-Teitelboim 1997
Duality invariance of the action
Gravitational electric-magnetic The action is invariant under rotations in the (1,2) plane of the duality
Marc Henneaux vector potentials (“electric-magnetic duality rotations") because
Introduction ²ab and δab are invariant tensors.
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality To conclude : the “proof" using the standard form of the em duality transformations that the second order Maxwell action S[A ] 1 R d4xF Fµν is not invariant under duality µ = − 4 µν transformations is simply incorrect because it is based on a form of the duality transformations that is inconsistent with the existence of the dynamical variable Aµ. A consistent set of transformations leaves the action invariant. Deser-Teitelboim 1976, Deser-Gomberoff-Henneaux-Teitelboim 1997
Duality invariance of the action
Gravitational electric-magnetic The action is invariant under rotations in the (1,2) plane of the duality
Marc Henneaux vector potentials (“electric-magnetic duality rotations") because
Introduction ²ab and δab are invariant tensors.
Electromagnetism The action is also invariant under the gauge transformations, in D 4 = Gravitational a a a duality in D 4 ~A ~A ~ Λ . (linearized = −→ + ∇ gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Deser-Teitelboim 1976, Deser-Gomberoff-Henneaux-Teitelboim 1997
Duality invariance of the action
Gravitational electric-magnetic The action is invariant under rotations in the (1,2) plane of the duality
Marc Henneaux vector potentials (“electric-magnetic duality rotations") because
Introduction ²ab and δab are invariant tensors.
Electromagnetism The action is also invariant under the gauge transformations, in D 4 = Gravitational a a a duality in D 4 ~A ~A ~ Λ . (linearized = −→ + ∇ gravity)
Twisted self-duality for p-forms To conclude : the “proof" using the standard form of the em
Twisted duality transformations that the second order Maxwell action self-duality for 1 R 4 µν (linearized) S[Aµ] 4 d xFµνF is not invariant under duality Gravity = − transformations is simply incorrect because it is based on a Duality invariance and spacetime form of the duality transformations that is inconsistent with covariance the existence of the dynamical variable A . A consistent set of Conclusions µ transformations leaves the action invariant.
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action
Gravitational electric-magnetic The action is invariant under rotations in the (1,2) plane of the duality
Marc Henneaux vector potentials (“electric-magnetic duality rotations") because
Introduction ²ab and δab are invariant tensors.
Electromagnetism The action is also invariant under the gauge transformations, in D 4 = Gravitational a a a duality in D 4 ~A ~A ~ Λ . (linearized = −→ + ∇ gravity)
Twisted self-duality for p-forms To conclude : the “proof" using the standard form of the em
Twisted duality transformations that the second order Maxwell action self-duality for 1 R 4 µν (linearized) S[Aµ] 4 d xFµνF is not invariant under duality Gravity = − transformations is simply incorrect because it is based on a Duality invariance and spacetime form of the duality transformations that is inconsistent with covariance the existence of the dynamical variable A . A consistent set of Conclusions µ transformations leaves the action invariant. Deser-Teitelboim 1976, Deser-Gomberoff-Henneaux-Teitelboim 1997
Marc Henneaux Gravitational electric-magnetic duality The introduction of the second potential makes the formalism local. The analysis can be extended to several abelian vector fields (U(n)-duality invariance), as well as to vector fields appropriately coupled to scalar fields (Sp(n,R)-duality invariance), with the same conclusions. Bunster-Henneaux 2011. In this formulation, Poincaré invariance is not manifest, however. More on this later.
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Bunster-Henneaux 2011. In this formulation, Poincaré invariance is not manifest, however. More on this later.
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism The introduction of the second potential makes the formalism in D 4 = local. Gravitational duality in D 4 The analysis can be extended to several abelian vector fields (linearized = gravity) (U(n)-duality invariance), as well as to vector fields appropriately Twisted R self-duality for coupled to scalar fields (Sp(n, )-duality invariance), with the p-forms same conclusions. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality In this formulation, Poincaré invariance is not manifest, however. More on this later.
Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism The introduction of the second potential makes the formalism in D 4 = local. Gravitational duality in D 4 The analysis can be extended to several abelian vector fields (linearized = gravity) (U(n)-duality invariance), as well as to vector fields appropriately Twisted R self-duality for coupled to scalar fields (Sp(n, )-duality invariance), with the p-forms same conclusions. Twisted self-duality for Bunster-Henneaux 2011. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism The introduction of the second potential makes the formalism in D 4 = local. Gravitational duality in D 4 The analysis can be extended to several abelian vector fields (linearized = gravity) (U(n)-duality invariance), as well as to vector fields appropriately Twisted R self-duality for coupled to scalar fields (Sp(n, )-duality invariance), with the p-forms same conclusions. Twisted self-duality for Bunster-Henneaux 2011. (linearized) Gravity In this formulation, Poincaré invariance is not manifest, however. Duality invariance and spacetime More on this later. covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic The Riemann tensor duality Marc Henneaux 1 ¡ ¢ Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ Introduction = −2 − − +
Electromagnetism in D 4 fulfills the identity = Gravitational Rλ[µρσ] 0. duality in D 4 = (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic The Riemann tensor duality Marc Henneaux 1 ¡ ¢ Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ Introduction = −2 − − +
Electromagnetism in D 4 fulfills the identity = Gravitational Rλ[µρσ] 0. duality in D 4 = (linearized = gravity)
Twisted The Einstein equations are Rµν 0. self-duality for = p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality also fulfills ∗R 0, ∗R 0 λ[µρσ] = µν =
and conversely.
Duality invariance of the Einstein equations
Gravitational electric-magnetic The Riemann tensor duality Marc Henneaux 1 ¡ ¢ Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ Introduction = −2 − − +
Electromagnetism in D 4 fulfills the identity = Gravitational Rλ[µρσ] 0. duality in D 4 = (linearized = gravity)
Twisted The Einstein equations are Rµν 0. self-duality for = p-forms This implies that the dual Riemann tensor
Twisted self-duality for 1 αβ (linearized) ∗Rλµρσ ²λµαβR ρσ Gravity = 2 Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality and conversely.
Duality invariance of the Einstein equations
Gravitational electric-magnetic The Riemann tensor duality Marc Henneaux 1 ¡ ¢ Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ Introduction = −2 − − +
Electromagnetism in D 4 fulfills the identity = Gravitational Rλ[µρσ] 0. duality in D 4 = (linearized = gravity)
Twisted The Einstein equations are Rµν 0. self-duality for = p-forms This implies that the dual Riemann tensor
Twisted self-duality for 1 αβ (linearized) ∗Rλµρσ ²λµαβR ρσ Gravity = 2 Duality invariance and spacetime covariance also fulfills Conclusions ∗R 0, ∗R 0 λ[µρσ] = µν =
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations
Gravitational electric-magnetic The Riemann tensor duality Marc Henneaux 1 ¡ ¢ Rλµρσ ∂λ∂ρhµσ ∂µ∂ρhλσ ∂λ∂σhµρ ∂µ∂σhλρ Introduction = −2 − − +
Electromagnetism in D 4 fulfills the identity = Gravitational Rλ[µρσ] 0. duality in D 4 = (linearized = gravity)
Twisted The Einstein equations are Rµν 0. self-duality for = p-forms This implies that the dual Riemann tensor
Twisted self-duality for 1 αβ (linearized) ∗Rλµρσ ²λµαβR ρσ Gravity = 2 Duality invariance and spacetime covariance also fulfills Conclusions ∗R 0, ∗R 0 λ[µρσ] = µν =
and conversely.
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism R cosαR sinα ∗R in D 4 → − = Gravitational ∗R sinαR cosα ∗R, duality in D 4 → + (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism R cosαR sinα ∗R in D 4 → − = Gravitational ∗R sinαR cosα ∗R, duality in D 4 → + (linearized = gravity)
Twisted or in (3 1)- fashion, self-duality for + p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism R cosαR sinα ∗R in D 4 → − = Gravitational ∗R sinαR cosα ∗R, duality in D 4 → + (linearized = gravity)
Twisted or in (3 1)- fashion, self-duality for + p-forms ij ij ij Twisted E cosαE sinαB self-duality for → − (linearized) ij ij ij Gravity B sinαE cosαB → + Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.
Duality invariance of the Einstein equations
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism R cosαR sinα ∗R in D 4 → − = Gravitational ∗R sinαR cosα ∗R, duality in D 4 → + (linearized = gravity)
Twisted or in (3 1)- fashion, self-duality for + p-forms ij ij ij Twisted E cosαE sinαB self-duality for → − (linearized) ij ij ij Gravity B sinαE cosαB → + Duality invariance and spacetime covariance where ij and ij are the electric and magnetic components of Conclusions E B the Riemann tensor, respectively.
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations
Gravitational electric-magnetic It follows that the Einstein equations are invariant under the duality
Marc Henneaux duality rotations
Introduction
Electromagnetism R cosαR sinα ∗R in D 4 → − = Gravitational ∗R sinαR cosα ∗R, duality in D 4 → + (linearized = gravity)
Twisted or in (3 1)- fashion, self-duality for + p-forms ij ij ij Twisted E cosαE sinαB self-duality for → − (linearized) ij ij ij Gravity B sinαE cosαB → + Duality invariance and spacetime covariance where ij and ij are the electric and magnetic components of Conclusions E B the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N.
Marc Henneaux Gravitational electric-magnetic duality 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. We exhibit right away the manifestly duality-invariant form of the action. It is obtained by starting from the first-order (Hamiltonian) action and solving the constraints.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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. We exhibit right away the manifestly duality-invariant form of the action. It is obtained by starting from the first-order (Hamiltonian) action and solving the constraints.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux Is this also a symmetry of the Pauli-Fierz action ? Introduction
Electromagnetism S[hµν] in D 4 = =Z Gravitational 1 4 £ ρ µν µν ρ µ ν µ ¤ duality in D 4 d x ∂ h ∂ρhµν 2∂µh ∂ρh ν 2∂ h∂ hµν ∂ h∂µh . (linearized = − 4 − + − gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The answer to the question turns out to be positive, just as for Maxwell’s theory. We exhibit right away the manifestly duality-invariant form of the action. It is obtained by starting from the first-order (Hamiltonian) action and solving the constraints.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux Is this also a symmetry of the Pauli-Fierz action ? Introduction
Electromagnetism S[hµν] in D 4 = =Z Gravitational 1 4 £ ρ µν µν ρ µ ν µ ¤ duality in D 4 d x ∂ h ∂ρhµν 2∂µh ∂ρh ν 2∂ h∂ hµν ∂ h∂µh . (linearized = − 4 − + − gravity) Twisted Note that the action is not expressed in terms of the curvature. self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality We exhibit right away the manifestly duality-invariant form of the action. It is obtained by starting from the first-order (Hamiltonian) action and solving the constraints.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux Is this also a symmetry of the Pauli-Fierz action ? Introduction
Electromagnetism S[hµν] in D 4 = =Z Gravitational 1 4 £ ρ µν µν ρ µ ν µ ¤ duality in D 4 d x ∂ h ∂ρhµν 2∂µh ∂ρh ν 2∂ h∂ hµν ∂ h∂µh . (linearized = − 4 − + − gravity) Twisted Note that the action is not expressed in terms of the curvature. self-duality for p-forms The answer to the question turns out to be positive, just as for Twisted self-duality for Maxwell’s theory. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality It is obtained by starting from the first-order (Hamiltonian) action and solving the constraints.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux Is this also a symmetry of the Pauli-Fierz action ? Introduction
Electromagnetism S[hµν] in D 4 = =Z Gravitational 1 4 £ ρ µν µν ρ µ ν µ ¤ duality in D 4 d x ∂ h ∂ρhµν 2∂µh ∂ρh ν 2∂ h∂ hµν ∂ h∂µh . (linearized = − 4 − + − gravity) Twisted Note that the action is not expressed in terms of the curvature. self-duality for p-forms The answer to the question turns out to be positive, just as for Twisted self-duality for Maxwell’s theory. (linearized) Gravity We exhibit right away the manifestly duality-invariant form of the Duality invariance and spacetime action. covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux Is this also a symmetry of the Pauli-Fierz action ? Introduction
Electromagnetism S[hµν] in D 4 = =Z Gravitational 1 4 £ ρ µν µν ρ µ ν µ ¤ duality in D 4 d x ∂ h ∂ρhµν 2∂µh ∂ρh ν 2∂ h∂ hµν ∂ h∂µh . (linearized = − 4 − + − gravity) Twisted Note that the action is not expressed in terms of the curvature. self-duality for p-forms The answer to the question turns out to be positive, just as for Twisted self-duality for Maxwell’s theory. (linearized) Gravity We exhibit right away the manifestly duality-invariant form of the Duality invariance and spacetime action. covariance It is obtained by starting from the first-order (Hamiltonian) Conclusions action and solving the constraints.
Marc Henneaux Gravitational electric-magnetic duality This step 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 The solution of the Hamiltonian constraint leads to the other 2 prepotential Zij. a Both prepotentials Zij are symmetric tensors (Young symmetry type ). Both are invariant under
δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality For instance, the momentum constraint ∂ πij 0 is solved by i = πij ²ipq²jrs∂ ∂ Z1 . = p r qs The solution of the Hamiltonian constraint leads to the other 2 prepotential Zij. a Both prepotentials Zij are symmetric tensors (Young symmetry type ). Both are invariant under
δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality This step introduces two prepotentials, one for the metric and Marc Henneaux one for its conjugate momentum. Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The solution of the Hamiltonian constraint leads to the other 2 prepotential Zij. a Both prepotentials Zij are symmetric tensors (Young symmetry type ). Both are invariant under
δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality This step introduces two prepotentials, one for the metric and Marc Henneaux one for its conjugate momentum. Introduction ij Electromagnetism For instance, the momentum constraint ∂iπ 0 is solved by in D 4 = = Gravitational ij ipq jrs Z1 . duality in D 4 π ² ² ∂p∂r qs (linearized = = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality a Both prepotentials Zij are symmetric tensors (Young symmetry type ). Both are invariant under
δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality This step introduces two prepotentials, one for the metric and Marc Henneaux one for its conjugate momentum. Introduction ij Electromagnetism For instance, the momentum constraint ∂iπ 0 is solved by in D 4 = = Gravitational ij ipq jrs Z1 . duality in D 4 π ² ² ∂p∂r qs (linearized = = gravity) The solution of the Hamiltonian constraint leads to the other Twisted 2 self-duality for prepotential Z . p-forms ij
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Both are invariant under
δZa ∂ ξa ∂ ξa 2²aδ ij = i j + j i + ij
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality This step introduces two prepotentials, one for the metric and Marc Henneaux one for its conjugate momentum. Introduction ij Electromagnetism For instance, the momentum constraint ∂iπ 0 is solved by in D 4 = = Gravitational ij ipq jrs Z1 . duality in D 4 π ² ² ∂p∂r qs (linearized = = gravity) The solution of the Hamiltonian constraint leads to the other Twisted 2 self-duality for prepotential Z . p-forms ij a Twisted Both prepotentials Z are symmetric tensors (Young symmetry self-duality for ij (linearized) Gravity type ).
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality This step introduces two prepotentials, one for the metric and Marc Henneaux one for its conjugate momentum. Introduction ij Electromagnetism For instance, the momentum constraint ∂iπ 0 is solved by in D 4 = = Gravitational ij ipq jrs Z1 . duality in D 4 π ² ² ∂p∂r qs (linearized = = gravity) The solution of the Hamiltonian constraint leads to the other Twisted 2 self-duality for prepotential Z . p-forms ij a Twisted Both prepotentials Z are symmetric tensors (Young symmetry self-duality for ij (linearized) Gravity type ). Duality invariance Both are invariant under and spacetime covariance a a a a Conclusions δZ ∂ ξ ∂ ξ 2² δ ij = i j + j i + ij
Marc Henneaux Gravitational electric-magnetic duality In terms of the prepotentials, the action reads
Z · Z ¸ ij S[Za ] dt 2 d3x²abD Z˙ H mn = − a bij −
where D ij D ij[Z ] is the co-Cotton tensor constructed out of a ≡ a the prepotential Zaij, and where the Hamiltonian is given by
Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a Here, Rij is the Ricci tensor constructed out of the prepotential a Zij.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Z · Z ¸ ij S[Za ] dt 2 d3x²abD Z˙ H mn = − a bij −
where D ij D ij[Z ] is the co-Cotton tensor constructed out of a ≡ a the prepotential Zaij, and where the Hamiltonian is given by
Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a Here, Rij is the Ricci tensor constructed out of the prepotential a Zij.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux In terms of the prepotentials, the action reads
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality where D ij D ij[Z ] is the co-Cotton tensor constructed out of a ≡ a the prepotential Zaij, and where the Hamiltonian is given by
Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a Here, Rij is the Ricci tensor constructed out of the prepotential a Zij.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux In terms of the prepotentials, the action reads
Introduction Z · Z ¸ Electromagnetism ij in D 4 a 3 ab ˙ = S[Z mn] dt 2 d x² Da Zbij H Gravitational = − − duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality and where the Hamiltonian is given by
Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a Here, Rij is the Ricci tensor constructed out of the prepotential a Zij.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux In terms of the prepotentials, the action reads
Introduction Z · Z ¸ Electromagnetism ij in D 4 a 3 ab ˙ = S[Z mn] dt 2 d x² Da Zbij H Gravitational = − − duality in D 4 = (linearized where D ij D ij[Z ] is the co-Cotton tensor constructed out of gravity) a ≡ a Twisted the prepotential Zaij, self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Z µ 3 ¶ H d3x 4Ra Rbij RaRb δ . = ij − 2 ab a Here, Rij is the Ricci tensor constructed out of the prepotential a Zij.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux In terms of the prepotentials, the action reads
Introduction Z · Z ¸ Electromagnetism ij in D 4 a 3 ab ˙ = S[Z mn] dt 2 d x² Da Zbij H Gravitational = − − duality in D 4 = (linearized where D ij D ij[Z ] is the co-Cotton tensor constructed out of gravity) a ≡ a Twisted the prepotential Zaij, self-duality for p-forms and where the Hamiltonian is given by Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality Marc Henneaux In terms of the prepotentials, the action reads
Introduction Z · Z ¸ Electromagnetism ij in D 4 a 3 ab ˙ = S[Z mn] dt 2 d x² Da Zbij H Gravitational = − − duality in D 4 = (linearized where D ij D ij[Z ] is the co-Cotton tensor constructed out of gravity) a ≡ a Twisted the prepotential Zaij, self-duality for p-forms and where the Hamiltonian is given by Twisted self-duality for (linearized) Z µ ¶ Gravity 3 a bij 3 a b H d x 4RijR R R δab. Duality invariance = − 2 and spacetime covariance a Here, Rij is the Ricci tensor constructed out of the prepotential Conclusions a Zij.
Marc Henneaux Gravitational electric-magnetic duality The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is also immediate, but one loses manifest space-time covariance. Just as for the Maxwell theory, there is a tension between manifest duality invariance and manifest space-time covariance. Henneaux-Teitelboim 2005.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Invariance under the gauge symmetries of the prepotentials is also immediate, but one loses manifest space-time covariance. Just as for the Maxwell theory, there is a tension between manifest duality invariance and manifest space-time covariance. Henneaux-Teitelboim 2005.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction Electromagnetism The action is manifestly invariant under duality rotations of the in D 4 = prepotentials. Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality but one loses manifest space-time covariance. Just as for the Maxwell theory, there is a tension between manifest duality invariance and manifest space-time covariance. Henneaux-Teitelboim 2005.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction Electromagnetism The action is manifestly invariant under duality rotations of the in D 4 = prepotentials. Gravitational duality in D 4 (linearized = Invariance under the gauge symmetries of the prepotentials is gravity) also immediate, Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Just as for the Maxwell theory, there is a tension between manifest duality invariance and manifest space-time covariance. Henneaux-Teitelboim 2005.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction Electromagnetism The action is manifestly invariant under duality rotations of the in D 4 = prepotentials. Gravitational duality in D 4 (linearized = Invariance under the gauge symmetries of the prepotentials is gravity) also immediate, Twisted self-duality for p-forms but one loses manifest space-time covariance.
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Henneaux-Teitelboim 2005.
Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction Electromagnetism The action is manifestly invariant under duality rotations of the in D 4 = prepotentials. Gravitational duality in D 4 (linearized = Invariance under the gauge symmetries of the prepotentials is gravity) also immediate, Twisted self-duality for p-forms but one loses manifest space-time covariance. Twisted Just as for the Maxwell theory, there is a tension between self-duality for (linearized) manifest duality invariance and manifest space-time covariance. Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Pauli-Fierz action
Gravitational electric-magnetic duality
Marc Henneaux
Introduction Electromagnetism The action is manifestly invariant under duality rotations of the in D 4 = prepotentials. Gravitational duality in D 4 (linearized = Invariance under the gauge symmetries of the prepotentials is gravity) also immediate, Twisted self-duality for p-forms but one loses manifest space-time covariance. Twisted Just as for the Maxwell theory, there is a tension between self-duality for (linearized) manifest duality invariance and manifest space-time covariance. Gravity
Duality invariance Henneaux-Teitelboim 2005. and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality In higher dimensions, 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. Cremmer-Julia 1978. We start with the p-form case.
Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality This is true for either electromagnetism or gravity. The duality-symmetric formulation is then based on the “twisted self-duality" reformulation of the theory. Cremmer-Julia 1978. We start with the p-form case.
Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 In higher dimensions, the curvature and its dual are tensors of = Gravitational different types. duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The duality-symmetric formulation is then based on the “twisted self-duality" reformulation of the theory. Cremmer-Julia 1978. We start with the p-form case.
Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 In higher dimensions, the curvature and its dual are tensors of = Gravitational different types. duality in D 4 (linearized = gravity) This is true for either electromagnetism or gravity.
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Cremmer-Julia 1978. We start with the p-form case.
Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 In higher dimensions, the curvature and its dual are tensors of = Gravitational different types. duality in D 4 (linearized = gravity) This is true for either electromagnetism or gravity. Twisted The duality-symmetric formulation is then based on the “twisted self-duality for p-forms self-duality" reformulation of the theory. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality We start with the p-form case.
Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 In higher dimensions, the curvature and its dual are tensors of = Gravitational different types. duality in D 4 (linearized = gravity) This is true for either electromagnetism or gravity. Twisted The duality-symmetric formulation is then based on the “twisted self-duality for p-forms self-duality" reformulation of the theory. Twisted self-duality for Cremmer-Julia 1978. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 In higher dimensions, the curvature and its dual are tensors of = Gravitational different types. duality in D 4 (linearized = gravity) This is true for either electromagnetism or gravity. Twisted The duality-symmetric formulation is then based on the “twisted self-duality for p-forms self-duality" reformulation of the theory. Twisted self-duality for Cremmer-Julia 1978. (linearized) Gravity We start with the p-form case. Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The Maxwell action Z µ 1 ¶ D λ1 λp 1 S[Aλ1 λp ] d x Fλ1 λp 1 F ··· + , ··· = − 2(p 1)! ··· + + with F dA, gives the equations of motion =
d ∗F 0. = On the other hand, it follows from the definition of F that
dF 0. =
The general solution to the equation of motion is ∗F dB for = some (D p 2)-form B. One calls the original form A the − − “electric potential" and the form B the “magnetic potential".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality On the other hand, it follows from the definition of F that
dF 0. =
The general solution to the equation of motion is ∗F dB for = some (D p 2)-form B. One calls the original form A the − − “electric potential" and the form B the “magnetic potential".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality
Marc Henneaux The Maxwell action
Introduction Z µ 1 ¶ D λ1 λp 1 Electromagnetism S[Aλ1 λp ] d x Fλ1 λp 1 F ··· + , in D 4 ··· = − 2(p 1)! ··· + = + Gravitational duality in D 4 with F dA, gives the equations of motion (linearized = = gravity) Twisted d ∗F 0. self-duality for = p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The general solution to the equation of motion is ∗F dB for = some (D p 2)-form B. One calls the original form A the − − “electric potential" and the form B the “magnetic potential".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality
Marc Henneaux The Maxwell action
Introduction Z µ 1 ¶ D λ1 λp 1 Electromagnetism S[Aλ1 λp ] d x Fλ1 λp 1 F ··· + , in D 4 ··· = − 2(p 1)! ··· + = + Gravitational duality in D 4 with F dA, gives the equations of motion (linearized = = gravity) Twisted d ∗F 0. self-duality for = p-forms
Twisted On the other hand, it follows from the definition of F that self-duality for (linearized) Gravity dF 0. Duality invariance = and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for p-forms
Gravitational electric-magnetic duality
Marc Henneaux The Maxwell action
Introduction Z µ 1 ¶ D λ1 λp 1 Electromagnetism S[Aλ1 λp ] d x Fλ1 λp 1 F ··· + , in D 4 ··· = − 2(p 1)! ··· + = + Gravitational duality in D 4 with F dA, gives the equations of motion (linearized = = gravity) Twisted d ∗F 0. self-duality for = p-forms
Twisted On the other hand, it follows from the definition of F that self-duality for (linearized) Gravity dF 0. Duality invariance = and spacetime covariance The general solution to the equation of motion is ∗F dB for Conclusions = some (D p 2)-form B. One calls the original form A the − − “electric potential" and the form B the “magnetic potential".
Marc Henneaux Gravitational electric-magnetic duality Conversely, assume that one has a p-form A and a (D p 2)-form B, the curvatures F dA and H dB of which are − − = = dual to one another,
(p 1)(D 1) 1 ∗F H,( 1) + − − ∗H F, = − = then A fulfills the Maxwell equations. In matrix form, one can rewrite the equations as
F S ∗F , = where µ ¶ µ (p 1)(D 1) 1¶ F 0 ( 1) + − − F , S − . = H = 1 0 One refers to these equations as the twisted self-dual formulation of Maxwell’s equations because S introduces a “twist".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality then A fulfills the Maxwell equations. In matrix form, one can rewrite the equations as
F S ∗F , = where µ ¶ µ (p 1)(D 1) 1¶ F 0 ( 1) + − − F , S − . = H = 1 0 One refers to these equations as the twisted self-dual formulation of Maxwell’s equations because S introduces a “twist".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality Conversely, assume that one has a p-form A and a Marc Henneaux (D p 2)-form B, the curvatures F dA and H dB of which are − − = = Introduction dual to one another, Electromagnetism in D 4 = F H,( 1)(p 1)(D 1) 1 H F, Gravitational ∗ + − − ∗ duality in D 4 = − = (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality In matrix form, one can rewrite the equations as
F S ∗F , = where µ ¶ µ (p 1)(D 1) 1¶ F 0 ( 1) + − − F , S − . = H = 1 0 One refers to these equations as the twisted self-dual formulation of Maxwell’s equations because S introduces a “twist".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality Conversely, assume that one has a p-form A and a Marc Henneaux (D p 2)-form B, the curvatures F dA and H dB of which are − − = = Introduction dual to one another, Electromagnetism in D 4 = F H,( 1)(p 1)(D 1) 1 H F, Gravitational ∗ + − − ∗ duality in D 4 = − = (linearized = gravity) then A fulfills the Maxwell equations.
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality where µ ¶ µ (p 1)(D 1) 1¶ F 0 ( 1) + − − F , S − . = H = 1 0 One refers to these equations as the twisted self-dual formulation of Maxwell’s equations because S introduces a “twist".
Twisted self-duality for p-forms
Gravitational electric-magnetic duality Conversely, assume that one has a p-form A and a Marc Henneaux (D p 2)-form B, the curvatures F dA and H dB of which are − − = = Introduction dual to one another, Electromagnetism in D 4 = F H,( 1)(p 1)(D 1) 1 H F, Gravitational ∗ + − − ∗ duality in D 4 = − = (linearized = gravity) then A fulfills the Maxwell equations.
Twisted self-duality for In matrix form, one can rewrite the equations as p-forms
Twisted self-duality for F S ∗F , (linearized) = Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for p-forms
Gravitational electric-magnetic duality Conversely, assume that one has a p-form A and a Marc Henneaux (D p 2)-form B, the curvatures F dA and H dB of which are − − = = Introduction dual to one another, Electromagnetism in D 4 = F H,( 1)(p 1)(D 1) 1 H F, Gravitational ∗ + − − ∗ duality in D 4 = − = (linearized = gravity) then A fulfills the Maxwell equations.
Twisted self-duality for In matrix form, one can rewrite the equations as p-forms
Twisted self-duality for F S ∗F , (linearized) = Gravity where Duality invariance µ ¶ µ (p 1)(D 1) 1¶ and spacetime F 0 ( 1) + − − covariance F , S − . = H = 1 0 Conclusions One refers to these equations as the twisted self-dual formulation of Maxwell’s equations because S introduces a “twist".
Marc Henneaux Gravitational electric-magnetic duality In the original variational principle, A and B do not play a symmetric role (only A appears). Can one formulate the dynamics in such a way that A and B are on equal footing ? This seems necessary in order to exhibit the hidden symmetries, where duality symmetry is built in. The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, Bunster-Henneaux). While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant).
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Can one formulate the dynamics in such a way that A and B are on equal footing ? This seems necessary in order to exhibit the hidden symmetries, where duality symmetry is built in. The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, Bunster-Henneaux). While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant).
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction In the original variational principle, A and B do not play a Electromagnetism in D 4 symmetric role (only A appears). = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality This seems necessary in order to exhibit the hidden symmetries, where duality symmetry is built in. The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, Bunster-Henneaux). While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant).
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction In the original variational principle, A and B do not play a Electromagnetism in D 4 symmetric role (only A appears). = Gravitational Can one formulate the dynamics in such a way that A and B are duality in D 4 (linearized = on equal footing ? gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, Bunster-Henneaux). While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant).
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction In the original variational principle, A and B do not play a Electromagnetism in D 4 symmetric role (only A appears). = Gravitational Can one formulate the dynamics in such a way that A and B are duality in D 4 (linearized = on equal footing ? gravity)
Twisted This seems necessary in order to exhibit the hidden symmetries, self-duality for p-forms where duality symmetry is built in.
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant).
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction In the original variational principle, A and B do not play a Electromagnetism in D 4 symmetric role (only A appears). = Gravitational Can one formulate the dynamics in such a way that A and B are duality in D 4 (linearized = on equal footing ? gravity)
Twisted This seems necessary in order to exhibit the hidden symmetries, self-duality for p-forms where duality symmetry is built in.
Twisted self-duality for The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, (linearized) Gravity Bunster-Henneaux).
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction In the original variational principle, A and B do not play a Electromagnetism in D 4 symmetric role (only A appears). = Gravitational Can one formulate the dynamics in such a way that A and B are duality in D 4 (linearized = on equal footing ? gravity)
Twisted This seems necessary in order to exhibit the hidden symmetries, self-duality for p-forms where duality symmetry is built in.
Twisted self-duality for The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, (linearized) Gravity Bunster-Henneaux). Duality invariance While the corresponding action is duality-symmetric, it lacks and spacetime covariance manifest space-time covariance (but of course it is covariant). Conclusions
Marc Henneaux Gravitational electric-magnetic duality The duality symmetric action reads
à ! Z ²k1 kpj1 jD p 1 D ··· ··· − − ˙ S[Ak1 kp ,Bj1 jD p 2 ] d x Hj1 jD p 1 Ak1 kp H ··· ··· − − = p!(D p 1)! ··· − − ··· − − − with 1 µ 1 ¶ j1 jD p 1 H Hj1 jD p 1 H ··· − − = 2 (D p 1)! ··· − − − − 1 µ 1 ¶ j1 jp 1 F ··· + Fj1 jp 1 +2 (p 1)! ··· + +
The equations of motion are that the electric field of A (respectively, of B) is equal to the magnetic field of B (respectively, of A), and so the electric energy of A is the magnetic energy of B. A and B are not only duality conjugate, but also canonically conjugate.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality à ! Z ²k1 kpj1 jD p 1 D ··· ··· − − ˙ S[Ak1 kp ,Bj1 jD p 2 ] d x Hj1 jD p 1 Ak1 kp H ··· ··· − − = p!(D p 1)! ··· − − ··· − − − with 1 µ 1 ¶ j1 jD p 1 H Hj1 jD p 1 H ··· − − = 2 (D p 1)! ··· − − − − 1 µ 1 ¶ j1 jp 1 F ··· + Fj1 jp 1 +2 (p 1)! ··· + +
The equations of motion are that the electric field of A (respectively, of B) is equal to the magnetic field of B (respectively, of A), and so the electric energy of A is the magnetic energy of B. A and B are not only duality conjugate, but also canonically conjugate.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic The duality symmetric action reads duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The equations of motion are that the electric field of A (respectively, of B) is equal to the magnetic field of B (respectively, of A), and so the electric energy of A is the magnetic energy of B. A and B are not only duality conjugate, but also canonically conjugate.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic The duality symmetric action reads duality
Marc Henneaux à ! Introduction Z ²k1 kpj1 jD p 1 D ··· ··· − − ˙ Electromagnetism S[Ak1 kp ,Bj1 jD p 2 ] d x Hj1 jD p 1 Ak1 kp H in D 4 ··· ··· − − = p!(D p 1)! ··· − − ··· − = − − Gravitational duality in D 4 with (linearized = gravity) µ ¶ 1 1 j j Twisted 1 D p 1 H Hj1 jD p 1 H ··· − − self-duality for = 2 (D p 1)! ··· − − p-forms − − 1 µ 1 ¶ Twisted j1 jp 1 self-duality for F ··· + Fj1 jp 1 (linearized) +2 (p 1)! ··· + Gravity +
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality A and B are not only duality conjugate, but also canonically conjugate.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic The duality symmetric action reads duality
Marc Henneaux à ! Introduction Z ²k1 kpj1 jD p 1 D ··· ··· − − ˙ Electromagnetism S[Ak1 kp ,Bj1 jD p 2 ] d x Hj1 jD p 1 Ak1 kp H in D 4 ··· ··· − − = p!(D p 1)! ··· − − ··· − = − − Gravitational duality in D 4 with (linearized = gravity) µ ¶ 1 1 j j Twisted 1 D p 1 H Hj1 jD p 1 H ··· − − self-duality for = 2 (D p 1)! ··· − − p-forms − − 1 µ 1 ¶ Twisted j1 jp 1 self-duality for F ··· + Fj1 jp 1 (linearized) +2 (p 1)! ··· + Gravity +
Duality invariance and spacetime The equations of motion are that the electric field of A covariance
Conclusions (respectively, of B) is equal to the magnetic field of B (respectively, of A), and so the electric energy of A is the magnetic energy of B.
Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of p-forms
Gravitational electric-magnetic The duality symmetric action reads duality
Marc Henneaux à ! Introduction Z ²k1 kpj1 jD p 1 D ··· ··· − − ˙ Electromagnetism S[Ak1 kp ,Bj1 jD p 2 ] d x Hj1 jD p 1 Ak1 kp H in D 4 ··· ··· − − = p!(D p 1)! ··· − − ··· − = − − Gravitational duality in D 4 with (linearized = gravity) µ ¶ 1 1 j j Twisted 1 D p 1 H Hj1 jD p 1 H ··· − − self-duality for = 2 (D p 1)! ··· − − p-forms − − 1 µ 1 ¶ Twisted j1 jp 1 self-duality for F ··· + Fj1 jp 1 (linearized) +2 (p 1)! ··· + Gravity +
Duality invariance and spacetime The equations of motion are that the electric field of A covariance
Conclusions (respectively, of B) is equal to the magnetic field of B (respectively, of A), and so the electric energy of A is the magnetic energy of B. A and B are not only duality conjugate, but also canonically conjugate.
Marc Henneaux Gravitational electric-magnetic duality The method covers also p-form interactions. The recipe is always the same : (i) Write the original (non-duality symmetric) action in Hamiltonian form ; (ii) The conjugate momenta (electric field components) are constrained by Gauss’ law resulting from gauge invariance, e.g. ∂ πk 0 for electromagnetism. Solve Gauss’ law constraint to k = express the electric field in terms of the dual potential B, k kij1 jD 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the = ··· − action. Spacetime covariance is guaranteed by the fact that the energy-momentum components obey the Dirac-Schwinger commutation relations. Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The recipe is always the same : (i) Write the original (non-duality symmetric) action in Hamiltonian form ; (ii) The conjugate momenta (electric field components) are constrained by Gauss’ law resulting from gauge invariance, e.g. ∂ πk 0 for electromagnetism. Solve Gauss’ law constraint to k = express the electric field in terms of the dual potential B, k kij1 jD 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the = ··· − action. Spacetime covariance is guaranteed by the fact that the energy-momentum components obey the Dirac-Schwinger commutation relations. Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality (i) Write the original (non-duality symmetric) action in Hamiltonian form ; (ii) The conjugate momenta (electric field components) are constrained by Gauss’ law resulting from gauge invariance, e.g. ∂ πk 0 for electromagnetism. Solve Gauss’ law constraint to k = express the electric field in terms of the dual potential B, k kij1 jD 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the = ··· − action. Spacetime covariance is guaranteed by the fact that the energy-momentum components obey the Dirac-Schwinger commutation relations. Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux The recipe is always the same : Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality (ii) The conjugate momenta (electric field components) are constrained by Gauss’ law resulting from gauge invariance, e.g. ∂ πk 0 for electromagnetism. Solve Gauss’ law constraint to k = express the electric field in terms of the dual potential B, k kij1 jD 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the = ··· − action. Spacetime covariance is guaranteed by the fact that the energy-momentum components obey the Dirac-Schwinger commutation relations. Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux The recipe is always the same : Introduction
Electromagnetism (i) Write the original (non-duality symmetric) action in in D 4 = Hamiltonian form ; Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Spacetime covariance is guaranteed by the fact that the energy-momentum components obey the Dirac-Schwinger commutation relations. Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux The recipe is always the same : Introduction
Electromagnetism (i) Write the original (non-duality symmetric) action in in D 4 = Hamiltonian form ; Gravitational duality in D 4 (linearized = (ii) The conjugate momenta (electric field components) are gravity) constrained by Gauss’ law resulting from gauge invariance, e.g. Twisted k self-duality for ∂kπ 0 for electromagnetism. Solve Gauss’ law constraint to p-forms = express the electric field in terms of the dual potential B, Twisted k kij j self-duality for 1 D 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the (linearized) = ··· − Gravity action.
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Gauge invariance is manifest.
Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux The recipe is always the same : Introduction
Electromagnetism (i) Write the original (non-duality symmetric) action in in D 4 = Hamiltonian form ; Gravitational duality in D 4 (linearized = (ii) The conjugate momenta (electric field components) are gravity) constrained by Gauss’ law resulting from gauge invariance, e.g. Twisted k self-duality for ∂kπ 0 for electromagnetism. Solve Gauss’ law constraint to p-forms = express the electric field in terms of the dual potential B, Twisted k kij j self-duality for 1 D 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the (linearized) = ··· − Gravity action.
Duality invariance and spacetime Spacetime covariance is guaranteed by the fact that the covariance energy-momentum components obey the Dirac-Schwinger Conclusions commutation relations.
Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of p-forms
Gravitational electric-magnetic duality The method covers also p-form interactions. Marc Henneaux The recipe is always the same : Introduction
Electromagnetism (i) Write the original (non-duality symmetric) action in in D 4 = Hamiltonian form ; Gravitational duality in D 4 (linearized = (ii) The conjugate momenta (electric field components) are gravity) constrained by Gauss’ law resulting from gauge invariance, e.g. Twisted k self-duality for ∂kπ 0 for electromagnetism. Solve Gauss’ law constraint to p-forms = express the electric field in terms of the dual potential B, Twisted k kij j self-duality for 1 D 3 (π ² ··· − ∂iBj1 jD 3 for em) and insert the result in the (linearized) = ··· − Gravity action.
Duality invariance and spacetime Spacetime covariance is guaranteed by the fact that the covariance energy-momentum components obey the Dirac-Schwinger Conclusions commutation relations. Gauge invariance is manifest.
Marc Henneaux Gravitational electric-magnetic duality Again, only understood for linearized gravity. For definiteness, consider D 5. In that case, the “dual graviton" = is described by a tensor Tαβγ of mixed symmetry type described by the Young tableau
T T , T 0. αβγ = [αβ]γ [αβγ] = The theory of a massless tensor field of this Young symmetry type has been constructed by Curtright, who wrote the action. The gauge symmetries are
δT 2∂ σ 2∂ α 2∂ α α1α2β = [α1 α2]β + [α1 α2]β − β α1α2
Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality For definiteness, consider D 5. In that case, the “dual graviton" = is described by a tensor Tαβγ of mixed symmetry type described by the Young tableau
T T , T 0. αβγ = [αβ]γ [αβγ] = The theory of a massless tensor field of this Young symmetry type has been constructed by Curtright, who wrote the action. The gauge symmetries are
δT 2∂ σ 2∂ α 2∂ α α1α2β = [α1 α2]β + [α1 α2]β − β α1α2
Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux Again, only understood for linearized gravity.
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The theory of a massless tensor field of this Young symmetry type has been constructed by Curtright, who wrote the action. The gauge symmetries are
δT 2∂ σ 2∂ α 2∂ α α1α2β = [α1 α2]β + [α1 α2]β − β α1α2
Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux Again, only understood for linearized gravity.
Introduction For definiteness, consider D 5. In that case, the “dual graviton" = Electromagnetism is described by a tensor Tαβγ of mixed symmetry type described in D 4 = by the Young tableau Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for Tαβγ T[αβ]γ, T[αβγ] 0. p-forms = =
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The gauge symmetries are
δT 2∂ σ 2∂ α 2∂ α α1α2β = [α1 α2]β + [α1 α2]β − β α1α2
Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux Again, only understood for linearized gravity.
Introduction For definiteness, consider D 5. In that case, the “dual graviton" = Electromagnetism is described by a tensor Tαβγ of mixed symmetry type described in D 4 = by the Young tableau Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for Tαβγ T[αβ]γ, T[αβγ] 0. p-forms = =
Twisted The theory of a massless tensor field of this Young symmetry type self-duality for (linearized) has been constructed by Curtright, who wrote the action. Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux Again, only understood for linearized gravity.
Introduction For definiteness, consider D 5. In that case, the “dual graviton" = Electromagnetism is described by a tensor Tαβγ of mixed symmetry type described in D 4 = by the Young tableau Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for Tαβγ T[αβ]γ, T[αβγ] 0. p-forms = =
Twisted The theory of a massless tensor field of this Young symmetry type self-duality for (linearized) has been constructed by Curtright, who wrote the action. Gravity The gauge symmetries are Duality invariance and spacetime covariance δTα1α2β 2∂[α1 σα2]β 2∂[α1 αα2]β 2∂βαα1α2 Conclusions = + −
Marc Henneaux Gravitational electric-magnetic duality The gauge invariant curvature is E 6∂ T , α1α2α3β1β2 = [α1 α2α3][β1,β2] which is a tensor of Young symmetry type
The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities ∂ E 0, E 0 [β0 β1β2β3]ρ1ρ2 = β1β2β3[ρ1ρ2,ρ3] = These identities imply in turn the existence of Tαβγ. The equations of motion are
E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality which is a tensor of Young symmetry type
The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities ∂ E 0, E 0 [β0 β1β2β3]ρ1ρ2 = β1β2β3[ρ1ρ2,ρ3] = These identities imply in turn the existence of Tαβγ. The equations of motion are
E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Twisted self-duality for gravity
Gravitational electric-magnetic duality The gauge invariant curvature is Eα1α2α3β1β2 6∂[α1 Tα2α3][β1,β2], Marc Henneaux =
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities ∂ E 0, E 0 [β0 β1β2β3]ρ1ρ2 = β1β2β3[ρ1ρ2,ρ3] = These identities imply in turn the existence of Tαβγ. The equations of motion are
E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Twisted self-duality for gravity
Gravitational electric-magnetic duality The gauge invariant curvature is Eα1α2α3β1β2 6∂[α1 Tα2α3][β1,β2], Marc Henneaux = which is a tensor of Young symmetry type Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality These identities imply in turn the existence of Tαβγ. The equations of motion are
E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Twisted self-duality for gravity
Gravitational electric-magnetic duality The gauge invariant curvature is Eα1α2α3β1β2 6∂[α1 Tα2α3][β1,β2], Marc Henneaux = which is a tensor of Young symmetry type Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities
Twisted ∂[β0 Eβ1β2β3]ρ1ρ2 0, Eβ1β2β3[ρ1ρ2,ρ3] 0 self-duality for = = (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The equations of motion are
E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Twisted self-duality for gravity
Gravitational electric-magnetic duality The gauge invariant curvature is Eα1α2α3β1β2 6∂[α1 Tα2α3][β1,β2], Marc Henneaux = which is a tensor of Young symmetry type Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities
Twisted ∂[β0 Eβ1β2β3]ρ1ρ2 0, Eβ1β2β3[ρ1ρ2,ρ3] 0 self-duality for = = (linearized) These identities imply in turn the existence of T . Gravity αβγ
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for gravity
Gravitational electric-magnetic duality The gauge invariant curvature is Eα1α2α3β1β2 6∂[α1 Tα2α3][β1,β2], Marc Henneaux = which is a tensor of Young symmetry type Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms The tensor Eβ1β2β3ρ1ρ2 obeys the differential “Bianchi" identities
Twisted ∂[β0 Eβ1β2β3]ρ1ρ2 0, Eβ1β2β3[ρ1ρ2,ρ3] 0 self-duality for = = (linearized) These identities imply in turn the existence of T . Gravity αβγ
Duality invariance The equations of motion are and spacetime covariance
Conclusions E 0 α1α2β =
for the “Ricci tensor" Eα1α2β.
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for gravity
Gravitational electric-magnetic duality
Marc Henneaux The Einstein equations Rµν 0 for the Riemann tensor Rµναβ[h] imply Introduction = that the dual Riemann tensor Eβ1β2β3ρ1ρ2 , defined by Electromagnetism in D 4 = Gravitational 1 α1α2 duality in D 4 Eβ1β2β3ρ1ρ2 ²β1β2β3α1α2 R ρ1ρ2 (linearized = = 2! gravity) 1 β1β2β3 Twisted Rα1α2ρ1ρ2 ²α1α2β1β2β3 E ρ1ρ2 self-duality for = − 3! p-forms Twisted is of Young symmetry type self-duality for (linearized) Gravity . Duality invariance and spacetime covariance
Conclusions Here, hαβ is the spin-2 (Pauli-Fierz) field.
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Furthermore, (i) the tensor Eβ1β2β3ρ1ρ2 obeys the differential identities Gravitational duality in D 4 ∂[β0 Eβ1β2β3]ρ1ρ2 0, Eβ1β2β3[ρ1ρ2,ρ3] 0 that guarantee the existence of (linearized = = = gravity) a tensor Tαβµ such that
Twisted self-duality for E E [T]; p-forms β1β2β3ρ1ρ2 = β1β2β3ρ1ρ2 Twisted self-duality for (linearized) and (ii) the field equations for the dual tensor Tαβµ are satisfied. Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Conversely, one may reformulate the gravitational field equations as twisted self-duality equations as follows.
Let hµν and Tαβµ be tensor fields of respective Young symmetry
types and , and let Rα1α2ρ1ρ2 [h] and Eβ1β2β3ρ1ρ2 [T] be the corresponding gauge-invariant curvatures.
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Let hµν and Tαβµ be tensor fields of respective Young symmetry
types and , and let Rα1α2ρ1ρ2 [h] and Eβ1β2β3ρ1ρ2 [T] be the corresponding gauge-invariant curvatures.
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational Conversely, one may reformulate the gravitational field equations duality in D 4 (linearized = as twisted self-duality equations as follows. gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational Conversely, one may reformulate the gravitational field equations duality in D 4 (linearized = as twisted self-duality equations as follows. gravity) Let h and T be tensor fields of respective Young symmetry Twisted µν αβµ self-duality for p-forms types and , and let Rα1α2ρ1ρ2 [h] and Eβ1β2β3ρ1ρ2 [T] be Twisted self-duality for the corresponding gauge-invariant curvatures. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The “twisted self-duality conditions", which express that E is the dual of R (we drop indices)
R ∗E, E ∗R, = − = or, in matrix notations, R S ∗R, = with µR¶ µ0 1¶ R , S − , = E = 1 0
imply that hµν and Tαβµ are both solutions of the linearized Einstein equations and the Curtright equations,
R 0, E 0. µν = µνα =
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality imply that hµν and Tαβµ are both solutions of the linearized Einstein equations and the Curtright equations,
R 0, E 0. µν = µνα =
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality The “twisted self-duality conditions", which express that E is the Marc Henneaux dual of R (we drop indices) Introduction Electromagnetism R ∗E, E ∗R, in D 4 = = − = Gravitational duality in D 4 or, in matrix notations, (linearized = gravity) R S ∗R, = Twisted self-duality for with p-forms µR¶ µ0 1¶ Twisted R , S − , self-duality for = E = 1 0 (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality The “twisted self-duality conditions", which express that E is the Marc Henneaux dual of R (we drop indices) Introduction Electromagnetism R ∗E, E ∗R, in D 4 = = − = Gravitational duality in D 4 or, in matrix notations, (linearized = gravity) R S ∗R, = Twisted self-duality for with p-forms µR¶ µ0 1¶ Twisted R , S − , self-duality for = E = 1 0 (linearized) Gravity imply that hµν and Tαβµ are both solutions of the linearized Duality invariance and spacetime Einstein equations and the Curtright equations, covariance Conclusions R 0, E 0. µν = µνα =
Marc Henneaux Gravitational electric-magnetic duality This is because, as we have seen, the cyclic identity for E (respectively, for R) implies that the Ricci tensor of hαβ (respectively, of Tαβγ) vanishes. The above equations are called twisted self-duality conditions for linearized gravity because if one views the curvature R as a single object, then these conditions express that this object is self-dual up to a twist, given by the matrix S . The twisted self-duality equations put the graviton and its dual on an identical footing.
One can define electric and magnetic fields for hαβ and Tαβγ. The twisted self-duality conditions are equivalent to B [T] E [h], B [h] E [T]. ijrs = ijrs ijr = − ijr
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The above equations are called twisted self-duality conditions for linearized gravity because if one views the curvature R as a single object, then these conditions express that this object is self-dual up to a twist, given by the matrix S . The twisted self-duality equations put the graviton and its dual on an identical footing.
One can define electric and magnetic fields for hαβ and Tαβγ. The twisted self-duality conditions are equivalent to B [T] E [h], B [h] E [T]. ijrs = ijrs ijr = − ijr
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction This is because, as we have seen, the cyclic identity for E Electromagnetism (respectively, for R) implies that the Ricci tensor of hαβ in D 4 = (respectively, of Tαβγ) vanishes. Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality One can define electric and magnetic fields for hαβ and Tαβγ. The twisted self-duality conditions are equivalent to B [T] E [h], B [h] E [T]. ijrs = ijrs ijr = − ijr
Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction This is because, as we have seen, the cyclic identity for E Electromagnetism (respectively, for R) implies that the Ricci tensor of hαβ in D 4 = (respectively, of Tαβγ) vanishes. Gravitational duality in D 4 (linearized = The above equations are called twisted self-duality conditions for gravity) linearized gravity because if one views the curvature R as a single Twisted self-duality for object, then these conditions express that this object is self-dual p-forms up to a twist, given by the matrix S . The twisted self-duality Twisted self-duality for equations put the graviton and its dual on an identical footing. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality conditions for gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction This is because, as we have seen, the cyclic identity for E Electromagnetism (respectively, for R) implies that the Ricci tensor of hαβ in D 4 = (respectively, of Tαβγ) vanishes. Gravitational duality in D 4 (linearized = The above equations are called twisted self-duality conditions for gravity) linearized gravity because if one views the curvature R as a single Twisted self-duality for object, then these conditions express that this object is self-dual p-forms up to a twist, given by the matrix S . The twisted self-duality Twisted self-duality for equations put the graviton and its dual on an identical footing. (linearized) Gravity One can define electric and magnetic fields for hαβ and Tαβγ. The Duality invariance and spacetime twisted self-duality conditions are equivalent to covariance Bijrs[T] Eijrs[h], Bijr[h] Eijr[T]. Conclusions = = −
Marc Henneaux Gravitational electric-magnetic duality One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Although technically more involved, the procedure goes exactly as for p-forms. (i)Write the action in Hamiltonian form. (ii) Solve the constraints. This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Although technically more involved, the procedure goes exactly as for p-forms. (i)Write the action in Hamiltonian form. (ii) Solve the constraints. This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality (i)Write the action in Hamiltonian form. (ii) Solve the constraints. This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 Although technically more involved, the procedure goes exactly (linearized = gravity) as for p-forms.
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality (ii) Solve the constraints. This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 Although technically more involved, the procedure goes exactly (linearized = gravity) as for p-forms. Twisted (i)Write the action in Hamiltonian form. self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 Although technically more involved, the procedure goes exactly (linearized = gravity) as for p-forms. Twisted (i)Write the action in Hamiltonian form. self-duality for p-forms (ii) Solve the constraints. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality (iii) Insert the solution of the constraints back into the action.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 Although technically more involved, the procedure goes exactly (linearized = gravity) as for p-forms. Twisted (i)Write the action in Hamiltonian form. self-duality for p-forms (ii) Solve the constraints. Twisted self-duality for (linearized) This step introduces “prepotentials", of respective Young Gravity
Duality invariance symmetry type and , which are again canonically and spacetime covariance conjugate.
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux One can also derive the gravitational twisted self-duality Introduction equations from a variational principle where h and T are on the Electromagnetism in D 4 same footing. = Gravitational duality in D 4 Although technically more involved, the procedure goes exactly (linearized = gravity) as for p-forms. Twisted (i)Write the action in Hamiltonian form. self-duality for p-forms (ii) Solve the constraints. Twisted self-duality for (linearized) This step introduces “prepotentials", of respective Young Gravity
Duality invariance symmetry type and , which are again canonically and spacetime covariance conjugate. Conclusions (iii) Insert the solution of the constraints back into the action.
Marc Henneaux Gravitational electric-magnetic duality The end result is exactly the same, whether one starts from the Paui-Fierz or the Curtright action.
The prepotential is at the same time the prepotential for the ijk Pauli-Fierz field hij and for the momentum π conjugate to the Curtright field, while the prepotential is at the same time
the prepotential for the Curtright field Tijk and for the momentum πij conjugate to the Pauli-Fierz field. The equations of motion equate the electric field of h (respectively, T) with the magnetic field of T (respectively, h). The “electric energy" of one is the “magnetic energy" of the other. The details can be found in Bunster-Henneaux-Hörtner 2013.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The prepotential is at the same time the prepotential for the ijk Pauli-Fierz field hij and for the momentum π conjugate to the Curtright field, while the prepotential is at the same time
the prepotential for the Curtright field Tijk and for the momentum πij conjugate to the Pauli-Fierz field. The equations of motion equate the electric field of h (respectively, T) with the magnetic field of T (respectively, h). The “electric energy" of one is the “magnetic energy" of the other. The details can be found in Bunster-Henneaux-Hörtner 2013.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality Marc Henneaux The end result is exactly the same, whether one starts from the Introduction Paui-Fierz or the Curtright action. Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The equations of motion equate the electric field of h (respectively, T) with the magnetic field of T (respectively, h). The “electric energy" of one is the “magnetic energy" of the other. The details can be found in Bunster-Henneaux-Hörtner 2013.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality Marc Henneaux The end result is exactly the same, whether one starts from the Introduction Paui-Fierz or the Curtright action. Electromagnetism in D 4 = The prepotential is at the same time the prepotential for the Gravitational duality in D 4 ijk (linearized = Pauli-Fierz field hij and for the momentum π conjugate to the gravity)
Twisted Curtright field, while the prepotential is at the same time self-duality for p-forms the prepotential for the Curtright field Tijk and for the Twisted ij self-duality for momentum π conjugate to the Pauli-Fierz field. (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The “electric energy" of one is the “magnetic energy" of the other. The details can be found in Bunster-Henneaux-Hörtner 2013.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality Marc Henneaux The end result is exactly the same, whether one starts from the Introduction Paui-Fierz or the Curtright action. Electromagnetism in D 4 = The prepotential is at the same time the prepotential for the Gravitational duality in D 4 ijk (linearized = Pauli-Fierz field hij and for the momentum π conjugate to the gravity)
Twisted Curtright field, while the prepotential is at the same time self-duality for p-forms the prepotential for the Curtright field Tijk and for the Twisted ij self-duality for momentum π conjugate to the Pauli-Fierz field. (linearized) Gravity The equations of motion equate the electric field of h Duality invariance and spacetime (respectively, T) with the magnetic field of T (respectively, h). covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The details can be found in Bunster-Henneaux-Hörtner 2013.
Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality Marc Henneaux The end result is exactly the same, whether one starts from the Introduction Paui-Fierz or the Curtright action. Electromagnetism in D 4 = The prepotential is at the same time the prepotential for the Gravitational duality in D 4 ijk (linearized = Pauli-Fierz field hij and for the momentum π conjugate to the gravity)
Twisted Curtright field, while the prepotential is at the same time self-duality for p-forms the prepotential for the Curtright field Tijk and for the Twisted ij self-duality for momentum π conjugate to the Pauli-Fierz field. (linearized) Gravity The equations of motion equate the electric field of h Duality invariance and spacetime (respectively, T) with the magnetic field of T (respectively, h). covariance The “electric energy" of one is the “magnetic energy" of the other. Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of gravity
Gravitational electric-magnetic duality Marc Henneaux The end result is exactly the same, whether one starts from the Introduction Paui-Fierz or the Curtright action. Electromagnetism in D 4 = The prepotential is at the same time the prepotential for the Gravitational duality in D 4 ijk (linearized = Pauli-Fierz field hij and for the momentum π conjugate to the gravity)
Twisted Curtright field, while the prepotential is at the same time self-duality for p-forms the prepotential for the Curtright field Tijk and for the Twisted ij self-duality for momentum π conjugate to the Pauli-Fierz field. (linearized) Gravity The equations of motion equate the electric field of h Duality invariance and spacetime (respectively, T) with the magnetic field of T (respectively, h). covariance The “electric energy" of one is the “magnetic energy" of the other. Conclusions The details can be found in Bunster-Henneaux-Hörtner 2013.
Marc Henneaux Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest spacetime covariance. Does this tell us something ? Duality invariance might be more fundamental (Bunster-Henneaux 2013). One may indeed 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Does this tell us something ? Duality invariance might be more fundamental (Bunster-Henneaux 2013). One may indeed 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance might be more fundamental (Bunster-Henneaux 2013). One may indeed 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality One may indeed 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 Duality invariance might be more fundamental = Gravitational (Bunster-Henneaux 2013). duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 Duality invariance might be more fundamental = Gravitational (Bunster-Henneaux 2013). duality in D 4 (linearized = One may indeed show that in the simple case of an Abelian vector gravity) field (e.m.), duality invariance implies Poincaré invariance. Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality 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.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 Duality invariance might be more fundamental = Gravitational (Bunster-Henneaux 2013). duality in D 4 (linearized = One may indeed show that in the simple case of an Abelian vector gravity) field (e.m.), duality invariance implies Poincaré invariance. Twisted self-duality for p-forms The control of spacetime covariance is achieved through the
Twisted Dirac-Schwinger commutation relations for the self-duality for (linearized) energy-momentum tensor components. Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality and it implies Poincaré invariance.
Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 Duality invariance might be more fundamental = Gravitational (Bunster-Henneaux 2013). duality in D 4 (linearized = One may indeed show that in the simple case of an Abelian vector gravity) field (e.m.), duality invariance implies Poincaré invariance. Twisted self-duality for p-forms The control of spacetime covariance is achieved through the
Twisted Dirac-Schwinger commutation relations for the self-duality for (linearized) energy-momentum tensor components. Gravity
Duality invariance The commutation relation and spacetime covariance ij ¡ ¢ [H (x),H (x0)] δ Hi(x0) Hi(x) δ,j(x,x0) Conclusions = + is the only possibility for two conjugate transverse vectors~E,~B,
Marc Henneaux Gravitational electric-magnetic duality Duality invariance and spacetime covariance
Gravitational electric-magnetic duality Tension between manifest duality-invariance and manifest
Marc Henneaux spacetime covariance.
Introduction Does this tell us something ?
Electromagnetism in D 4 Duality invariance might be more fundamental = Gravitational (Bunster-Henneaux 2013). duality in D 4 (linearized = One may indeed show that in the simple case of an Abelian vector gravity) field (e.m.), duality invariance implies Poincaré invariance. Twisted self-duality for p-forms The control of spacetime covariance is achieved through the
Twisted Dirac-Schwinger commutation relations for the self-duality for (linearized) energy-momentum tensor components. Gravity
Duality invariance The commutation relation and spacetime covariance ij ¡ ¢ [H (x),H (x0)] δ Hi(x0) Hi(x) δ,j(x,x0) Conclusions = + is the only possibility for two conjugate transverse vectors~E,~B, and it implies Poincaré invariance.
Marc Henneaux Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. It is an off-shell symmetry (i.e., symmetry of the action and not just of the equations of motion). This implies the existence of a conserved (Noether) charge, and the fact that the symmetry is expected to hold at the quantum level. Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).
These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality It is an off-shell symmetry (i.e., symmetry of the action and not just of the equations of motion). This implies the existence of a conserved (Noether) charge, and the fact that the symmetry is expected to hold at the quantum level. Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).
These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality This implies the existence of a conserved (Noether) charge, and the fact that the symmetry is expected to hold at the quantum level. Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).
These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Manifestly duality invariant formulations do not exhibit manifest Poincaré invariance. Duality invariance might be more fundamental (Bunster-Henneaux 2013).
These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = This implies the existence of a conserved (Noether) charge, and Gravitational duality in D 4 the fact that the symmetry is expected to hold at the quantum (linearized = gravity) level.
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Duality invariance might be more fundamental (Bunster-Henneaux 2013).
These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = This implies the existence of a conserved (Noether) charge, and Gravitational duality in D 4 the fact that the symmetry is expected to hold at the quantum (linearized = gravity) level.
Twisted self-duality for Manifestly duality invariant formulations do not exhibit manifest p-forms Poincaré invariance. Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality These results are relevant for the E10-conjecture, since E10 has duality symmetry built in.
The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = This implies the existence of a conserved (Noether) charge, and Gravitational duality in D 4 the fact that the symmetry is expected to hold at the quantum (linearized = gravity) level.
Twisted self-duality for Manifestly duality invariant formulations do not exhibit manifest p-forms Poincaré invariance. Twisted self-duality for Duality invariance might be more fundamental (linearized) Gravity (Bunster-Henneaux 2013). Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = This implies the existence of a conserved (Noether) charge, and Gravitational duality in D 4 the fact that the symmetry is expected to hold at the quantum (linearized = gravity) level.
Twisted self-duality for Manifestly duality invariant formulations do not exhibit manifest p-forms Poincaré invariance. Twisted self-duality for Duality invariance might be more fundamental (linearized) Gravity (Bunster-Henneaux 2013). Duality invariance and spacetime These results are relevant for the E10-conjecture, since E10 has covariance duality symmetry built in. Conclusions
Marc Henneaux Gravitational electric-magnetic duality Conclusions
Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux It is an off-shell symmetry (i.e., symmetry of the action and not Introduction just of the equations of motion). Electromagnetism in D 4 = This implies the existence of a conserved (Noether) charge, and Gravitational duality in D 4 the fact that the symmetry is expected to hold at the quantum (linearized = gravity) level.
Twisted self-duality for Manifestly duality invariant formulations do not exhibit manifest p-forms Poincaré invariance. Twisted self-duality for Duality invariance might be more fundamental (linearized) Gravity (Bunster-Henneaux 2013). Duality invariance and spacetime These results are relevant for the E10-conjecture, since E10 has covariance duality symmetry built in. Conclusions The search for an E10-invariant action is legitimate, but this action might not be manifestly space-time covariant.
Marc Henneaux Gravitational electric-magnetic duality For p-forms, non-minimal couplings (as in supergravity) can be introduced without problem. For gravity, however, only the linearized theory has been dealt with so-far. Can one go beyond the linear level ? Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Other questions : Magnetic sources, asymptotic symmetries. Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality For gravity, however, only the linearized theory has been dealt with so-far. Can one go beyond the linear level ? Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Other questions : Magnetic sources, asymptotic symmetries. Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Other questions : Magnetic sources, asymptotic symmetries. Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational For gravity, however, only the linearized theory has been dealt duality in D 4 (linearized = with so-far. Can one go beyond the linear level ? gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Other questions : Magnetic sources, asymptotic symmetries. Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational For gravity, however, only the linearized theory has been dealt duality in D 4 (linearized = with so-far. Can one go beyond the linear level ? gravity)
Twisted Positive indications : Taub-NUT, Geroch group/ Ehlers group self-duality for p-forms upon dimensional reduction, cosmological billiards.
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Other questions : Magnetic sources, asymptotic symmetries. Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational For gravity, however, only the linearized theory has been dealt duality in D 4 (linearized = with so-far. Can one go beyond the linear level ? gravity)
Twisted Positive indications : Taub-NUT, Geroch group/ Ehlers group self-duality for p-forms upon dimensional reduction, cosmological billiards.
Twisted self-duality for The manifestly duality symmetric actions will most likely exhibit (linearized) Gravity some sort of non-locality.
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Much work remains to be done...
Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational For gravity, however, only the linearized theory has been dealt duality in D 4 (linearized = with so-far. Can one go beyond the linear level ? gravity)
Twisted Positive indications : Taub-NUT, Geroch group/ Ehlers group self-duality for p-forms upon dimensional reduction, cosmological billiards.
Twisted self-duality for The manifestly duality symmetric actions will most likely exhibit (linearized) Gravity some sort of non-locality. Duality invariance Other questions : Magnetic sources, asymptotic symmetries. and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Conclusions
Gravitational electric-magnetic duality
Marc Henneaux
Introduction For p-forms, non-minimal couplings (as in supergravity) can be Electromagnetism introduced without problem. in D 4 = Gravitational For gravity, however, only the linearized theory has been dealt duality in D 4 (linearized = with so-far. Can one go beyond the linear level ? gravity)
Twisted Positive indications : Taub-NUT, Geroch group/ Ehlers group self-duality for p-forms upon dimensional reduction, cosmological billiards.
Twisted self-duality for The manifestly duality symmetric actions will most likely exhibit (linearized) Gravity some sort of non-locality. Duality invariance Other questions : Magnetic sources, asymptotic symmetries. and spacetime covariance Much work remains to be done... Conclusions
Marc Henneaux Gravitational electric-magnetic duality Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity) THANK YOU ! Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality Gravitational electric-magnetic duality
Marc Henneaux
Introduction
Electromagnetism in D 4 = Gravitational duality in D 4 (linearized = gravity)
Twisted self-duality for p-forms
Twisted self-duality for (linearized) Gravity
Duality invariance and spacetime covariance
Conclusions
Marc Henneaux Gravitational electric-magnetic duality