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Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Gravitational electric-magnetic duality Marc Henneaux Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance November 2013 Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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). Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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, Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 ). Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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. Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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". Marc Henneaux Gravitational electric-magnetic duality 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 Introduction Gravitational electric-magnetic duality Marc Henneaux It has indeed been conjectured 10-15 years ago that the 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 Introduction Gravitational electric-magnetic duality Marc Henneaux It has indeed been conjectured 10-15 years ago that the 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 Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 6 7 Introduction Gravitational electric-magnetic duality Marc Henneaux It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms −1 0 1 2 3 4 5 or E11 Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 6 7 Introduction Gravitational electric-magnetic duality Marc Henneaux It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms 0 −1 1 2 3 4 5 6 7 or E11 11 Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance 10 9 8 7 6 5 4 Conclusions Marc Henneaux Gravitational electric-magnetic duality 3 2 1 Introduction Gravitational electric-magnetic duality Marc Henneaux It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10 Introduction 8 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms 0 −1 1 2 3 4 5 6 7 or E11 11 Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions 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. Marc Henneaux Gravitational electric-magnetic duality 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 Introduction Gravitational electric-magnetic duality Marc Henneaux One crucial feature of these algebras is that they treat 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 Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 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. 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 Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity D − 3 boxes                      Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity D − 3 boxes                      Duality invariance and spacetime covariance Conclusions simultaneously appear. Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity D − 3 boxes                      Duality invariance and spacetime covariance Conclusions simultaneously appear. Understanding gravitational duality is thus important in this context. Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : 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 Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) discuss first electric-magnetic duality in its original D = 4 electromagnetic context... Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms 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 ; Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity 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; Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : 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 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 ; Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux The purpose of this talk is to : 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 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. Marc Henneaux Gravitational electric-magnetic duality EM duality as an off-shell symmetry of the Maxwell theory in D = 4 Gravitational electric-magnetic duality Marc Henneaux We start with the Maxwell theory in 4 dimensions. 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 EM duality as an off-shell symmetry of the Maxwell theory in D = 4 Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 We start with the Maxwell theory in 4 dimensions. The duality transformations are usually written in terms of the field strengths 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 EM duality as an off-shell symmetry of the Maxwell theory in D = 4 Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms We start with the Maxwell theory in 4 dimensions. The duality transformations are usually written in terms of the field strengths F µν → cos α F µν − sin α ∗F µν ∗ µν F → sin α F µν + cos α ∗F µν , Twisted self-duality for (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 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 We start with the Maxwell theory in 4 dimensions. The duality transformations are usually written in terms of the field strengths F µν → cos α F µν − sin α ∗F µν ∗ µν F → sin α F µν + cos α ∗F µν , or in (3 + 1)- fashion, 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 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 We start with the Maxwell theory in 4 dimensions. The duality transformations are usually written in terms of the field strengths F µν → cos α F µν − sin α ∗F µν ∗ µν → sin α F µν + cos α ∗F µν , F or in (3 + 1)- fashion, E → cos α E + sin α B B → − sin α E + cos α B. Marc Henneaux Gravitational electric-magnetic duality 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 EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux These transformations are well known to leave the Maxwell 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 EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux These transformations are well known to leave the Maxwell equations dF = 0, d∗F = 0, or Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) ∇ · E = 0, ∇ · B = 0, ∂E ∂B − ∇ × B = 0, + ∇ × E = 0, ∂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 EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux These transformations are well known to leave the Maxwell equations dF = 0, d∗F = 0, or Introduction Electromagnetism in D = 4 ∇ · E = 0, ∇ · B = 0, ∂E ∂B − ∇ × B = 0, + ∇ × E = 0, ∂t ∂t Gravitational duality in D = 4 (linearized gravity) 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 EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux These transformations are well known to leave the Maxwell equations dF = 0, d∗F = 0, or Introduction Electromagnetism in D = 4 ∇ · E = 0, ∇ · B = 0, ∂E ∂B − ∇ × B = 0, + ∇ × E = 0, ∂t ∂t Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity invariant. We claim that these transformations also leave the Maxwell action 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 These transformations are well known to leave the Maxwell equations dF = 0, d∗F = 0, or Introduction Electromagnetism in D = 4 ∇ · E = 0, ∇ · B = 0, ∂E ∂B − ∇ × B = 0, + ∇ × E = 0, ∂t ∂t 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 invariant. We claim that these transformations also leave the Maxwell action S=− 1 4 Z d4 xFµν ∗F µν = Marc Henneaux 1 2 Z d4 x(E2 − B2 ) Gravitational electric-magnetic duality EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux These transformations are well known to leave the Maxwell equations dF = 0, d∗F = 0, or Introduction Electromagnetism in D = 4 ∇ · E = 0, ∇ · B = 0, ∂E ∂B − ∇ × B = 0, + ∇ × E = 0, ∂t ∂t Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity invariant. We claim that these transformations also leave the Maxwell action Duality invariance and spacetime covariance S=− Conclusions 1 4 Z d4 xFµν ∗F µν = 1 2 Z d4 x(E2 − B2 ) invariant. Marc Henneaux Gravitational electric-magnetic duality 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 EM duality as an off-shell symmetry Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 As a by-product, we shall also clarify why the hyperbolic rotations in the (E, B)-plane 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 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 As a by-product, we shall also clarify why the hyperbolic rotations in the (E, B)-plane E → cosh α E + sinh α B B → sinh α E + cosh α B, Twisted self-duality for (linearized) 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 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity As a by-product, we shall also clarify why the hyperbolic rotations in the (E, B)-plane E → cosh α E + sinh α B B → sinh α E + cosh α B, which seemingly leave the Maxwell action invariant, 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 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 As a by-product, we shall also clarify why the hyperbolic rotations in the (E, B)-plane E → cosh α E + sinh α B B → sinh α E + cosh α B, 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. Marc Henneaux Gravitational electric-magnetic duality 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 Duality transformations of the potential - The problem Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 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 , 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 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 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 Fµν = ∂µ Aν − ∂ν Aµ (so that B = ∇ × A). Twisted 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 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 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 Fµν = ∂µ Aν − ∂ν Aµ (so that B = ∇ × A). Furthermore, one must know these transformations off-shell since one must go off-shell to check invariance of the action. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Fµν = ∂µ Aν − ∂ν Aµ (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 ! Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Fµν = ∂µ Aν − ∂ν Aµ (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. Marc Henneaux Gravitational electric-magnetic duality 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 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 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) The proof is elementary. Once Aµ is introduced, dF = 0 is an identity. But dF 0 = 0 does not hold off-shell (unless α is a multiple of π but then F and its dual are not mixed), where F 0 is the new field strength cos α F −sin α ∗F after duality rotation. Hence there is no A0 such that F 0 = dA0 . 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 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 The proof is elementary. Once Aµ is introduced, dF = 0 is an identity. But dF 0 = 0 does not hold off-shell (unless α is a multiple of π but then F and its dual are not mixed), where F 0 is the new field strength cos α F −sin α ∗F after duality rotation. Hence there is no A0 such that F 0 = 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. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 The proof is elementary. Once Aµ is introduced, dF = 0 is an identity. But dF 0 = 0 does not hold off-shell (unless α is a multiple of π but then F and its dual are not mixed), where F 0 is the new field strength cos α F −sin α ∗F after duality rotation. Hence there is no A0 such that F 0 = 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. 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 (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 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 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. 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 (linearized gravity) Twisted self-duality for p-forms 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. 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 (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 Explicit form of the duality transformation of the vector potential under em duality Gravitational electric-magnetic duality Marc Henneaux Introduction Going from the transformations of the field strength to the transformations of the vector potential requires integrations and introduces non-local terms. 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 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 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 space, where the inverse Laplacian 4−1 can be given a meaning. Terms non local in time (and ä−1 ) are much more tricky. We also use the gauge ambiguity in the definition of δAµ in such a way that δA0 = 0. 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 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance 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 space, where the inverse Laplacian 4−1 can be given a meaning. Terms non local in time (and ä−1 ) are much more tricky. We also use the gauge ambiguity in the definition of δAµ in such a way that δA0 = 0. The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by ´ ³ δA0 = 0, δAi = −²4−1 ²ijk ∂j F0k . 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 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions 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 space, where the inverse Laplacian 4−1 can be given a meaning. Terms non local in time (and ä−1 ) are much more tricky. We also use the gauge ambiguity in the definition of δAµ in such a way that δA0 = 0. The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by ´ ³ δA0 = 0, δAi = −²4−1 ²ijk ∂j F0k . It implies δBi = −²E i − ²4−1 (∂i ∂j F0j ) = −²E i − ²4−1 (∂i ∂µ F0µ ), i.e., δBi = −²E i on-shell. 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 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions 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 space, where the inverse Laplacian 4−1 can be given a meaning. Terms non local in time (and ä−1 ) are much more tricky. We also use the gauge ambiguity in the definition of δAµ in such a way that δA0 = 0. The duality transformation of the vector potential is then given (up to a residual gauge symmetry) by ´ ³ δA0 = 0, δAi = −²4−1 ²ijk ∂j F0k . It implies δBi = −²E i − ²4−1 (∂i ∂j F0j ) = −²E i − ²4−1 (∂i ∂µ F0µ ), i.e., δBi = −²E i on-shell. Similarly, δEi = −δȦi = ²Bi − ²4−1 (²ijk ∂j ∂µ F µk ). Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the action Gravitational electric-magnetic duality Marc Henneaux Introduction The duality transformations of the vector-potential have the important property of leaving the Maxwell action invariant. 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 Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 d 1 δ d3 xE 2 = −² d3 xEi ²ijk 4−1 ∂j Ek 2 dt 2 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 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 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 d 1 δ d3 xE 2 = −² d3 xEi ²ijk 4−1 ∂j Ek 2 dt 2 and δ µ Z ¶ µ ¶ Z ¢ ¡ d 1 1 d3 xB2 = −² d3 xBi ²ijk 4−1 ∂j Bk 2 dt 2 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 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 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 d 1 δ d3 xE 2 = −² d3 xEi ²ijk 4−1 ∂j Ek 2 dt 2 and δ µ Z ¶ µ ¶ Z ¢ ¡ d 1 1 d3 xB2 = −² d3 xBi ²ijk 4−1 ∂j Bk 2 dt 2 ¡ ¢ R R so that δS = δ dtL = 0 (with L = 12 d3 x E 2 − B2 ). Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 d 1 δ d3 xE 2 = −² d3 xEi ²ijk 4−1 ∂j Ek 2 dt 2 and δ µ Z ¶ µ ¶ Z ¢ ¡ d 1 1 d3 xB2 = −² d3 xBi ²ijk 4−1 ∂j Bk 2 dt 2 ¡ ¢ R R so that δS = δ dtL = 0 (with L = 12 d3 x E 2 − B2 ). It is therefore a genuine Noether symmetry (with Noether charge etc). Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the action Gravitational electric-magnetic duality Marc Henneaux Introduction 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. 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 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 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 ≡~ B1 = ~ ∇ ×~ A1 , one introduces an additional vector potential ~ A2 through, ~ E ≡~ B2 = ~ ∇ ×~ A2 , one may rewrite the standard Maxwell action in terms of the two potentials Aa as Z ³ ´ 1 ˙b −δ ~ a ~b dx0 d3 x ²ab~ Ba · ~ A S= ab B · B . 2 Here, ²ab is given by ²ab = −²ba , ²12 = +1. Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the action Gravitational electric-magnetic duality Marc Henneaux Introduction 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. 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 Marc Henneaux Introduction Electromagnetism in D = 4 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 . 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 Marc Henneaux Introduction Electromagnetism in D = 4 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 . 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 To conclude : the “proof" using the standard form of the em duality transformations that the second order Maxwell action R S[Aµ ] = − 14 d4 xFµν F µν is not invariant under duality 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. Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the action Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 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 . 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 To conclude : the “proof" using the standard form of the em duality transformations that the second order Maxwell action R S[Aµ ] = − 14 d4 xFµν F µν is not invariant under duality 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 Marc Henneaux Gravitational electric-magnetic duality 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 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 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. 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 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 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. 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 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 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. Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 The Riemann tensor ¢ 1¡ Rλµρσ = − ∂λ ∂ρ hµσ − ∂µ ∂ρ hλσ − ∂λ ∂σ hµρ + ∂µ ∂σ hλρ 2 fulfills the identity Rλ[µρσ] = 0. 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 Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 The Riemann tensor ¢ 1¡ Rλµρσ = − ∂λ ∂ρ hµσ − ∂µ ∂ρ hλσ − ∂λ ∂σ hµρ + ∂µ ∂σ hλρ 2 fulfills the identity Rλ[µρσ] = 0. Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms The Einstein equations are Rµν = 0. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 The Riemann tensor ¢ 1¡ Rλµρσ = − ∂λ ∂ρ hµσ − ∂µ ∂ρ hλσ − ∂λ ∂σ hµρ + ∂µ ∂σ hλρ 2 fulfills the identity Rλ[µρσ] = 0. Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity The Einstein equations are Rµν = 0. This implies that the dual Riemann tensor ∗ 1 αβ Rλµρσ = ²λµαβ R ρσ 2 Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 The Riemann tensor ¢ 1¡ Rλµρσ = − ∂λ ∂ρ hµσ − ∂µ ∂ρ hλσ − ∂λ ∂σ hµρ + ∂µ ∂σ hλρ 2 fulfills the identity Rλ[µρσ] = 0. Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms The Einstein equations are Rµν = 0. This implies that the dual Riemann tensor Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions ∗ 1 αβ Rλµρσ = ²λµαβ R ρσ 2 also fulfills ∗ Rλ[µρσ] = 0, Marc Henneaux ∗ Rµν = 0 Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 The Riemann tensor ¢ 1¡ Rλµρσ = − ∂λ ∂ρ hµσ − ∂µ ∂ρ hλσ − ∂λ ∂σ hµρ + ∂µ ∂σ hλρ 2 fulfills the identity Rλ[µρσ] = 0. Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms The Einstein equations are Rµν = 0. This implies that the dual Riemann tensor Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance ∗ 1 αβ Rλµρσ = ²λµαβ R ρσ 2 also fulfills ∗ Conclusions Rλ[µρσ] = 0, ∗ Rµν = 0 and conversely. Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux It follows that the Einstein equations are invariant under the 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 Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) It follows that the Einstein equations are invariant under the duality rotations R → cos α R − sin α ∗R ∗ R → sin α R + cos α ∗R, 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 Einstein equations Gravitational electric-magnetic duality Marc Henneaux It follows that the Einstein equations are invariant under the duality rotations Introduction Electromagnetism in D = 4 R → cos α R − sin α ∗R ∗ R → sin α R + cos α ∗R, Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms or in (3 + 1)- fashion, Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux It follows that the Einstein equations are invariant under the duality rotations Introduction Electromagnetism in D = 4 R → cos α R − sin α ∗R ∗ R → sin α R + cos α ∗R, Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity or in (3 + 1)- fashion, E ij → cos α E ij − sin α B ij B ij → sin α E ij + cos α B ij Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux It follows that the Einstein equations are invariant under the duality rotations Introduction Electromagnetism in D = 4 R → cos α R − sin α ∗R ∗ R → sin α R + cos α ∗R, 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 or in (3 + 1)- fashion, E ij → cos α E ij − sin α B ij B ij → sin α E ij + cos α B ij where E ij and B ij are the electric and magnetic components of the Riemann tensor, respectively. Marc Henneaux Gravitational electric-magnetic duality Duality invariance of the Einstein equations Gravitational electric-magnetic duality Marc Henneaux It follows that the Einstein equations are invariant under the duality rotations Introduction Electromagnetism in D = 4 R → cos α R − sin α ∗R ∗ R → sin α R + cos α ∗R, 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 or in (3 + 1)- fashion, E ij → cos α E ij − sin α B ij B ij → sin α E ij + cos α B ij where E ij and B ij are the electric and magnetic components of the Riemann tensor, respectively. This transformation rotates the Schwarschild mass into the Taub-NUT parameter N. Marc Henneaux Gravitational electric-magnetic duality 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 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 in D = 4 Gravitational duality in D = 4 (linearized gravity) S[hµν ] = Z £ ¤ 1 ρ d4 x ∂ρ hµν ∂ρ hµν − 2∂µ hµν ∂ρ h ν + 2∂µ h∂ν hµν − ∂µ h∂µ h . − 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 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 in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms S[hµν ] = Z £ ¤ 1 ρ d4 x ∂ρ hµν ∂ρ hµν − 2∂µ hµν ∂ρ h ν + 2∂µ h∂ν hµν − ∂µ h∂µ h . − 4 Note that the action is not expressed in terms of the curvature. 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 Is this also a symmetry of the Pauli-Fierz action ? Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity S[hµν ] = Z £ ¤ 1 ρ d4 x ∂ρ hµν ∂ρ hµν − 2∂µ hµν ∂ρ h ν + 2∂µ h∂ν hµν − ∂µ h∂µ h . − 4 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. 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 Is this also a symmetry of the Pauli-Fierz action ? 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 S[hµν ] = Z £ ¤ 1 ρ d4 x ∂ρ hµν ∂ρ hµν − 2∂µ hµν ∂ρ h ν + 2∂µ h∂ν hµν − ∂µ h∂µ h . − 4 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. 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 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 S[hµν ] = Z £ ¤ 1 ρ d4 x ∂ρ hµν ∂ρ hµν − 2∂µ hµν ∂ρ h ν + 2∂µ h∂ν hµν − ∂µ h∂µ h . − 4 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. Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance of the Pauli-Fierz action Gravitational electric-magnetic duality Marc Henneaux This step introduces two prepotentials, one for the metric and 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 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) This step introduces two prepotentials, one for the metric and one for its conjugate momentum. For instance, the momentum constraint ∂i πij = 0 is solved by 1 . πij = ²ipq ²jrs ∂p ∂r Zqs 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 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 This step introduces two prepotentials, one for the metric and one for its conjugate momentum. For instance, the momentum constraint ∂i πij = 0 is solved by 1 . πij = ²ipq ²jrs ∂p ∂r Zqs The solution of the Hamiltonian constraint leads to the other prepotential Zij2 . 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 Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity This step introduces two prepotentials, one for the metric and one for its conjugate momentum. For instance, the momentum constraint ∂i πij = 0 is solved by 1 . πij = ²ipq ²jrs ∂p ∂r Zqs The solution of the Hamiltonian constraint leads to the other prepotential Zij2 . Both prepotentials Zija are symmetric tensors (Young symmetry type ). 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 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 This step introduces two prepotentials, one for the metric and one for its conjugate momentum. For instance, the momentum constraint ∂i πij = 0 is solved by 1 . πij = ²ipq ²jrs ∂p ∂r Zqs The solution of the Hamiltonian constraint leads to the other prepotential Zij2 . Both prepotentials Zija are symmetric tensors (Young symmetry type ). Both are invariant under δZ aij = ∂i ξaj + ∂j ξai + 2²a δij Marc Henneaux Gravitational electric-magnetic duality 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 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 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 S[Z amn ] = Z · dt −2 Z 3 ab d x² ij Da Żbij − 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 ¸ 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 S[Z amn ] = Z · dt −2 Z 3 ab d x² ij Da Żbij − H ij ¸ where Da ≡ D ij [Za ] is the co-Cotton tensor constructed out of the prepotential Zaij , 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 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms S[Z amn ] = Z · dt −2 Z 3 ab d x² ij Da Żbij − H ij ¸ where Da ≡ D ij [Za ] is the co-Cotton tensor constructed out of the prepotential Zaij , 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 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 S[Z amn ] = Z · dt −2 Z 3 ab d x² ij Da Żbij − H ¸ ij where Da ≡ D ij [Za ] is the co-Cotton tensor constructed out of the prepotential Zaij , and where the Hamiltonian is given by Z H= µ ¶ 3 d3 x 4Rija Rbij − Ra Rb δab . 2 Here, Rija is the Ricci tensor constructed out of the prepotential Zija . Marc Henneaux Gravitational electric-magnetic duality 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 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) The action is manifestly invariant under duality rotations of the prepotentials. 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 Pauli-Fierz action Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) The action is manifestly invariant under duality rotations of the prepotentials. Invariance under the gauge symmetries of the prepotentials is 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 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 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. 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 Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity 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. 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 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 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. Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 Twisted self-duality Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) In higher dimensions, the curvature and its dual are tensors of different types. 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 Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) In higher dimensions, the curvature and its dual are tensors of different types. 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 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 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. Twisted self-duality for (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 Gravitational duality in D = 4 (linearized gravity) In higher dimensions, the curvature and its dual are tensors of different types. This is true for either electromagnetism or gravity. Twisted self-duality for p-forms The duality-symmetric formulation is then based on the “twisted self-duality" reformulation of the theory. Twisted self-duality for (linearized) Gravity Cremmer-Julia 1978. 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 Gravitational duality in D = 4 (linearized gravity) In higher dimensions, the curvature and its dual are tensors of different types. This is true for either electromagnetism or gravity. Twisted self-duality for p-forms The duality-symmetric formulation is then based on the “twisted self-duality" reformulation of the theory. Twisted self-duality for (linearized) Gravity Cremmer-Julia 1978. We start with the p-form case. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 Twisted self-duality for p-forms Gravitational electric-magnetic duality Marc Henneaux The Maxwell action Z Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) S[Aλ1 ···λp ] = µ dD x − ¶ 1 Fλ1 ···λp+1 F λ1 ···λp+1 , 2(p + 1)! with F = dA, gives the equations of motion d ∗F = 0. 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 for p-forms Gravitational electric-magnetic duality Marc Henneaux The Maxwell action Z Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) S[Aλ1 ···λp ] = µ dD x − with F = dA, gives the equations of motion d ∗F = 0. Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity ¶ 1 Fλ1 ···λp+1 F λ1 ···λp+1 , 2(p + 1)! On the other hand, it follows from the definition of F that 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 Z Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) S[Aλ1 ···λp ] = µ dD x − with F = dA, gives the equations of motion d ∗F = 0. Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions ¶ 1 Fλ1 ···λp+1 F λ1 ···λp+1 , 2(p + 1)! 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". Marc Henneaux Gravitational electric-magnetic duality 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 Twisted self-duality for p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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, F = H, (−1)(p+1)(D−1)−1 ∗H = F, ∗ 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 for p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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, F = H, (−1)(p+1)(D−1)−1 ∗H = F, ∗ 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 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 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, F = H, (−1)(p+1)(D−1)−1 ∗H = F, ∗ then A fulfills the Maxwell equations. In matrix form, one can rewrite the equations as F = S ∗F , Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction 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, 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 F = H, (−1)(p+1)(D−1)−1 ∗H = F, ∗ then A fulfills the Maxwell equations. In matrix form, one can rewrite the equations as F = S ∗F , where F= µ ¶ µ F 0 , S = H 1 ¶ (−1)(p+1)(D−1)−1 . 0 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 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 Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 In the original variational principle, A and B do not play a 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 Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? 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-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 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. Twisted self-duality for (linearized) Gravity 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 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms This seems necessary in order to exhibit the hidden symmetries, where duality symmetry is built in. Twisted self-duality for (linearized) Gravity The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, 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 Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms This seems necessary in order to exhibit the hidden symmetries, where duality symmetry is built in. Twisted self-duality for (linearized) Gravity The answer is positive (Henneaux-Teitelboim, Sen-Schwarz, Bunster-Henneaux). Duality invariance and spacetime covariance While the corresponding action is duality-symmetric, it lacks manifest space-time covariance (but of course it is covariant). Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality The duality symmetric action reads 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 Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality The duality symmetric action reads Marc Henneaux Z Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity S[Ak1 ···kp , Bj1 ···jD−p−2 ] = Ã ²k1 ···kp j1 ···jD−p−1 Hj ···j Ȧk ···k − H d x p! (D − p − 1)! 1 D−p−1 1 p D with H = µ ¶ 1 1 Hj1 ···jD−p−1 H j1 ···jD−p−1 2 (D − p − 1)! µ ¶ 1 1 + F j1 ···jp+1 Fj1 ···jp+1 2 (p + 1)! Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality ! Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality The duality symmetric action reads Marc Henneaux Z 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 S[Ak1 ···kp , Bj1 ···jD−p−2 ] = Ã ²k1 ···kp j1 ···jD−p−1 Hj ···j Ȧk ···k − H d x p! (D − p − 1)! 1 D−p−1 1 p ! D with H = µ ¶ 1 1 Hj1 ···jD−p−1 H j1 ···jD−p−1 2 (D − p − 1)! µ ¶ 1 1 + F j1 ···jp+1 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. Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality The duality symmetric action reads Marc Henneaux Z 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 S[Ak1 ···kp , Bj1 ···jD−p−2 ] = Ã ²k1 ···kp j1 ···jD−p−1 Hj ···j Ȧk ···k − H d x p! (D − p − 1)! 1 D−p−1 1 p ! D with H = µ ¶ 1 1 Hj1 ···jD−p−1 H j1 ···jD−p−1 2 (D − p − 1)! µ ¶ 1 1 + F j1 ···jp+1 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. Marc Henneaux Gravitational electric-magnetic duality 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 Duality-symmetric formulation of p-forms Gravitational electric-magnetic duality Marc Henneaux The method covers also p-form interactions. 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-symmetric formulation of p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction The method covers also p-form interactions. The recipe is always the same : 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-symmetric formulation of p-forms Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) The method covers also p-form interactions. The recipe is always the same : (i) Write the original (non-duality symmetric) action in Hamiltonian form ; 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-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 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. ∂k πk = 0 for electromagnetism. Solve Gauss’ law constraint to express the electric field in terms of the dual potential B, (πk = ²kij1 ···jD−3 ∂i Bj1 ···jD−3 for em) and insert the result in the action. 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 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 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. ∂k πk = 0 for electromagnetism. Solve Gauss’ law constraint to express the electric field in terms of the dual potential B, (πk = ²kij1 ···jD−3 ∂i Bj1 ···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. Marc Henneaux Gravitational electric-magnetic duality 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 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. ∂k πk = 0 for electromagnetism. Solve Gauss’ law constraint to express the electric field in terms of the dual potential B, (πk = ²kij1 ···jD−3 ∂i Bj1 ···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. Marc Henneaux Gravitational electric-magnetic duality 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 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 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 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 Tαβγ = T[αβ]γ , T[αβγ] = 0. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Tαβγ = T[αβ]γ , T[αβγ] = 0. The theory of a massless tensor field of this Young symmetry type has been constructed by Curtright, who wrote the action. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Tαβγ = T[αβ]γ , T[αβγ] = 0. 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 δTα1 α2 β = 2∂[α1 σα2 ]β + 2∂[α1 αα2 ]β − 2∂β αα1 α2 Marc Henneaux Gravitational electric-magnetic duality 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 Twisted self-duality for gravity Gravitational electric-magnetic duality Marc Henneaux The gauge invariant curvature is Eα1 α2 α3 β1 β2 = 6∂[α1 Tα2 α3 ][β1 ,β2 ] , 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 Twisted self-duality for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction The gauge invariant curvature is Eα1 α2 α3 β1 β2 = 6∂[α1 Tα2 α3 ][β1 ,β2 ] , which is a tensor of Young symmetry type 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 Twisted self-duality for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction The gauge invariant curvature is Eα1 α2 α3 β1 β2 = 6∂[α1 Tα2 α3 ][β1 ,β2 ] , which is a tensor of Young symmetry type Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity The tensor Eβ1 β2 β3 ρ 1 ρ 2 obeys the differential “Bianchi" identities ∂[β0 Eβ1 β2 β3 ]ρ 1 ρ 2 = 0, Eβ1 β2 β3 [ρ 1 ρ 2 ,ρ 3 ] = 0 Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction The gauge invariant curvature is Eα1 α2 α3 β1 β2 = 6∂[α1 Tα2 α3 ][β1 ,β2 ] , which is a tensor of Young symmetry type Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity The tensor Eβ1 β2 β3 ρ 1 ρ 2 obeys the differential “Bianchi" identities ∂[β0 Eβ1 β2 β3 ]ρ 1 ρ 2 = 0, Eβ1 β2 β3 [ρ 1 ρ 2 ,ρ 3 ] = 0 These identities imply in turn the existence of Tαβγ . Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction The gauge invariant curvature is Eα1 α2 α3 β1 β2 = 6∂[α1 Tα2 α3 ][β1 ,β2 ] , which is a tensor of Young symmetry type 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 The tensor Eβ1 β2 β3 ρ 1 ρ 2 obeys the differential “Bianchi" identities ∂[β0 Eβ1 β2 β3 ]ρ 1 ρ 2 = 0, Eβ1 β2 β3 [ρ 1 ρ 2 ,ρ 3 ] = 0 These identities imply in turn the existence of Tαβγ . The equations of motion are Eα1 α2 β = 0 Conclusions for the “Ricci tensor" Eα1 α2 β . Marc Henneaux Gravitational electric-magnetic duality 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 The Einstein equations Rµν = 0 for the Riemann tensor Rµναβ [h] imply that the dual Riemann tensor Eβ1 β2 β3 ρ 1 ρ 2 , defined by Eβ1 β2 β3 ρ 1 ρ 2 = Rα1 α2 ρ 1 ρ 2 = is of Young symmetry type . Duality invariance and spacetime covariance Conclusions 1 α α ²β β β α α R 1 ρ21 ρ 2 2! 1 2 3 1 2 1 β β β − ²α1 α2 β1 β2 β3 E 1 2ρ 13ρ 2 3! 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 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Furthermore, (i) the tensor Eβ1 β2 β3 ρ 1 ρ 2 obeys the differential identities ∂[β0 Eβ1 β2 β3 ]ρ 1 ρ 2 = 0, Eβ1 β2 β3 [ρ 1 ρ 2 ,ρ 3 ] = 0 that guarantee the existence of a tensor Tαβµ such that Eβ1 β2 β3 ρ 1 ρ 2 = Eβ1 β2 β3 ρ 1 ρ 2 [T ]; and (ii) the field equations for the dual tensor Tαβµ are satisfied. 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 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 Twisted self-duality conditions for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Conversely, one may reformulate the gravitational field equations as twisted self-duality equations as follows. 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 duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity 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 Rα1 α2 ρ 1 ρ 2 [h] and Eβ1 β2 β3 ρ 1 ρ 2 [T ] be the corresponding gauge-invariant curvatures. 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 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 Twisted self-duality conditions for gravity Gravitational electric-magnetic duality Marc Henneaux The “twisted self-duality conditions", which express that E is the dual of R (we drop indices) Introduction R = − ∗ E, E = ∗ R, Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity or, in matrix notations, R = S ∗R, with R= µ ¶ µ R 0 , S = E 1 ¶ −1 , 0 Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Twisted self-duality conditions for gravity Gravitational electric-magnetic duality Marc Henneaux The “twisted self-duality conditions", which express that E is the dual of R (we drop indices) Introduction R = − ∗ E, E = ∗ R, 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 or, in matrix notations, R = S ∗R, with R= µ ¶ µ R 0 , S = E 1 ¶ −1 , 0 imply that hµν and Tαβµ are both solutions of the linearized Einstein equations and the Curtright equations, Conclusions Rµν = 0, Eµνα = 0. Marc Henneaux Gravitational electric-magnetic duality 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 Twisted self-duality conditions for gravity Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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. 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 duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity 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. 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 duality in D = 4 (linearized gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions 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 Bijrs [T ] = E ijrs [h], Bijr [h] = −E ijr [T ]. Marc Henneaux Gravitational electric-magnetic duality 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 Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the 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 Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Gravitational duality in D = 4 (linearized gravity) Although technically more involved, the procedure goes exactly as for p-forms. Introduction 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-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Gravitational duality in D = 4 (linearized gravity) Although technically more involved, the procedure goes exactly as for p-forms. Twisted self-duality for p-forms (i)Write the action in Hamiltonian form. Introduction Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Gravitational duality in D = 4 (linearized gravity) Although technically more involved, the procedure goes exactly as for p-forms. Twisted self-duality for p-forms (i)Write the action in Hamiltonian form. Introduction (ii) Solve the constraints. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Gravitational duality in D = 4 (linearized gravity) Although technically more involved, the procedure goes exactly as for p-forms. Twisted self-duality for p-forms (i)Write the action in Hamiltonian form. Introduction Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance (ii) Solve the constraints. This step introduces “prepotentials", of respective Young symmetry type and , which are again canonically conjugate. Conclusions Marc Henneaux Gravitational electric-magnetic duality Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Electromagnetism in D = 4 One can also derive the gravitational twisted self-duality equations from a variational principle where h and T are on the same footing. Gravitational duality in D = 4 (linearized gravity) Although technically more involved, the procedure goes exactly as for p-forms. Twisted self-duality for p-forms (i)Write the action in Hamiltonian form. Introduction Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions (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. Marc Henneaux Gravitational electric-magnetic duality 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 Duality-symmetric formulation of gravity Gravitational electric-magnetic duality Marc Henneaux Introduction The end result is exactly the same, whether one starts from the 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 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 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 Pauli-Fierz field hij and for the momentum πijk 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. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Pauli-Fierz field hij and for the momentum πijk 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). Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 Pauli-Fierz field hij and for the momentum πijk 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. Marc Henneaux Gravitational electric-magnetic duality 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 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 Pauli-Fierz field hij and for the momentum πijk 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. Marc Henneaux Gravitational electric-magnetic duality 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 Duality invariance and spacetime covariance Gravitational electric-magnetic duality Marc Henneaux Tension between manifest duality-invariance and manifest 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 and spacetime covariance Gravitational electric-magnetic duality Marc Henneaux Introduction Tension between manifest duality-invariance and manifest spacetime covariance. 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 Duality invariance and spacetime covariance Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) Tension between manifest duality-invariance and manifest spacetime covariance. Does this tell us something ? Duality invariance might be more fundamental (Bunster-Henneaux 2013). 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 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 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. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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 ¡ ¢ [H (x), H (x0 )] = δij H i (x0 ) + H i (x) δ,j (x, x0 ) 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 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 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 ¡ ¢ [H (x), H (x0 )] = δij H i (x0 ) + H i (x) δ,j (x, x0 ) is the only possibility for two conjugate transverse vectors ~ E, ~ B, and it implies Poincaré invariance. Marc Henneaux Gravitational electric-magnetic duality 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 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 Conclusions Gravitational electric-magnetic duality Electric-magnetic gravitational duality is a remarkable symmetry. Marc Henneaux Introduction Electromagnetism in D = 4 It is an off-shell symmetry (i.e., symmetry of the action and not just of the equations of motion). 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 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) 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. Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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). Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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. Marc Henneaux Gravitational electric-magnetic duality 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 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. Marc Henneaux Gravitational electric-magnetic duality 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 Conclusions Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 For p-forms, non-minimal couplings (as in supergravity) can be introduced without problem. 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 Conclusions Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 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 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. Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Conclusions Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. Twisted self-duality for (linearized) Gravity The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Conclusions Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. Twisted self-duality for (linearized) Gravity The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Duality invariance and spacetime covariance Other questions : Magnetic sources, asymptotic symmetries. Conclusions Marc Henneaux Gravitational electric-magnetic duality Conclusions Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D = 4 Gravitational duality in D = 4 (linearized gravity) 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 ? Twisted self-duality for p-forms Positive indications : Taub-NUT, Geroch group/ Ehlers group upon dimensional reduction, cosmological billiards. Twisted self-duality for (linearized) Gravity The manifestly duality symmetric actions will most likely exhibit some sort of non-locality. Duality invariance and spacetime covariance Other questions : Magnetic sources, asymptotic symmetries. 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) Twisted self-duality for p-forms THANK YOU ! 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

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