Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Introduction Electromagnetism Hidden Symmetries of Gravity and in D 4 Æ Gravitational Gravitational Duality duality in D 4 (linearized Æ gravity) Twisted self-duality Marc Henneaux Duality invariance and spacetime covariance Conclusions December 2013 1 / 34 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). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It is thought to be relevant to the so-called problem of “hidden symmetries". Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 2 / 34 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). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It is thought to be relevant to the so-called problem of “hidden symmetries". Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 2 / 34 Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It is thought to be relevant to the so-called problem of “hidden symmetries". Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Electromagnetism Originally considered in the context of electromagnetism, it also in D 4 Æ plays a key role in extended supergravity models, where the Gravitational duality in D 4 duality group (acting on the vector fields and the scalars) is (linearized Æ gravity) enlarged to U(n) or Sp(2n,R). Twisted self-duality Duality invariance and spacetime covariance Conclusions 2 / 34 It is thought to be relevant to the so-called problem of “hidden symmetries". Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Electromagnetism Originally considered in the context of electromagnetism, it also in D 4 Æ plays a key role in extended supergravity models, where the Gravitational duality in D 4 duality group (acting on the vector fields and the scalars) is (linearized Æ gravity) enlarged to U(n) or Sp(2n,R). Twisted self-duality Gravitational electric-magnetic duality (acting on the graviton) is Duality invariance also very intriguing. and spacetime covariance Conclusions 2 / 34 Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Electromagnetism Originally considered in the context of electromagnetism, it also in D 4 Æ plays a key role in extended supergravity models, where the Gravitational duality in D 4 duality group (acting on the vector fields and the scalars) is (linearized Æ gravity) enlarged to U(n) or Sp(2n,R). Twisted self-duality Gravitational electric-magnetic duality (acting on the graviton) is Duality invariance also very intriguing. and spacetime covariance It is thought to be relevant to the so-called problem of “hidden Conclusions symmetries". 2 / 34 It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 3 / 34 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hidden Symmetries of Gravity and Gravitational It has indeed been conjectured 10-15 years ago that the Duality infinite-dimensional Kac-Moody algebra E10 Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 3 / 34 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hidden Symmetries of Gravity and Gravitational It has indeed been conjectured 10-15 years ago that the Duality infinite-dimensional Kac-Moody algebra E10 Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 3 / 34 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hidden Symmetries of Gravity and Gravitational It has indeed been conjectured 10-15 years ago that the Duality infinite-dimensional Kac-Moody algebra E10 Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) or E11 Twisted self-duality Duality invariance and spacetime covariance Conclusions 3 / 34 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hidden Symmetries of Gravity and Gravitational It has indeed been conjectured 10-15 years ago that the Duality infinite-dimensional Kac-Moody algebra E10 Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) or E11 Twisted self-duality Duality invariance and spacetime covariance Conclusions 3 / 34 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 Introduction Hidden Symmetries of Gravity and Gravitational It has indeed been conjectured 10-15 years ago that the Duality infinite-dimensional Kac-Moody algebra E10 Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) or E11 Twisted self-duality Duality invariance and spacetime covariance Conclusions which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. 3 / 34 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 8 > > > > <> D 3 boxes ¡ > > > > :> simultaneously appear. Understanding gravitational duality is thus important in this context. Introduction Hidden Symmetries of Gravity and Gravitational Duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 4 / 34 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 8 > > > > <> D 3 boxes ¡ > > > > :> simultaneously appear. Understanding gravitational duality is thus important in this context. Introduction Hidden Symmetries of One crucial feature of these algebras is that they treat Gravity and Gravitational democratically all fields and their duals. Duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 4 / 34 Similarly, the graviton and its dual, described by a field with Young symmetry 8 > > > > <> D 3 boxes ¡ > > > > :> simultaneously appear. Understanding gravitational duality is thus important in this context. Introduction Hidden Symmetries of One crucial feature of these algebras is that they treat Gravity and Gravitational democratically all fields and their duals. Duality Marc Henneaux Whenever a (dynamical) p-form gauge field appears, its dual D p 2-form gauge field also appears. Introduction ¡ ¡ Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality Duality invariance and spacetime covariance Conclusions 4 / 34 8 > > > > <> D 3 boxes ¡ > > > > :> simultaneously appear. Understanding gravitational duality is thus important in this context. Introduction Hidden Symmetries of One crucial feature of these algebras is that they treat Gravity and Gravitational democratically all fields and their duals. Duality Marc Henneaux Whenever a (dynamical) p-form gauge field appears, its dual D p 2-form gauge field also appears. Introduction ¡ ¡ Electromagnetism Similarly, the graviton and its dual, described