Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational Gravitational electric-magnetic duality duality in D 4 (linearized Æ gravity) Twisted self-duality for Marc Henneaux p-forms Twisted self-duality for (linearized) Gravity Duality invariance November 2013 and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Electric-magnetic duality is a fascinating symmetry. Originally considered in the context of electromagnetism, it also plays a key role in extended supergravity models, where the duality group (acting on the vector fields and the scalars) is enlarged to U(n) or Sp(2n,R). In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Originally considered in the context of electromagnetism, it also plays a key role in extended supergravity models, where the duality group (acting on the vector fields and the scalars) is enlarged to U(n) or Sp(2n,R). In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality In these contexts, duality is a symmetry not only of the equations of motion, but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 Æ Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized Æ enlarged to U(n) or Sp(2n,R). gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality but also of the action (contrary to common belief). Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 Æ Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized Æ enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Gravitational electric-magnetic duality (acting on the graviton) is also very intriguing. It rotates for instance the Schwarzchild mass into the Taub-NUT parameter N. Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 Æ Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized Æ enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 Æ Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized Æ enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravitational electric-magnetic duality (acting on the graviton) is Gravity Duality invariance also very intriguing. It rotates for instance the Schwarzchild mass and spacetime covariance into the Taub-NUT parameter N. Conclusions Marc Henneaux Gravitational electric-magnetic duality Introduction Gravitational electric-magnetic duality Marc Henneaux Electric-magnetic duality is a fascinating symmetry. Introduction Originally considered in the context of electromagnetism, it also Electromagnetism plays a key role in extended supergravity models, where the in D 4 Æ Gravitational duality group (acting on the vector fields and the scalars) is duality in D 4 (linearized Æ enlarged to U(n) or Sp(2n,R). gravity) In these contexts, duality is a symmetry not only of the equations Twisted self-duality for of motion, p-forms Twisted but also of the action (contrary to common belief). self-duality for (linearized) Gravitational electric-magnetic duality (acting on the graviton) is Gravity Duality invariance also very intriguing. It rotates for instance the Schwarzchild mass and spacetime covariance into the Taub-NUT parameter N. Conclusions Gravitational duality is also thought to be relevant to the so-called problem of “hidden symmetries". Marc Henneaux Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the infinite-dimensional Kac-Moody algebra E10 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Gravitational electric-magnetic duality Marc Henneaux Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 which contains electric-magnetic gravitational duality, might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Gravitational electric-magnetic duality It has indeed been conjectured 10-15 years ago that the Marc Henneaux infinite-dimensional Kac-Moody algebra E10 Introduction Electromagnetism in D 4 Æ Gravitational duality in D 4 (linearized Æ gravity) Twisted self-duality for p-forms Twisted self-duality for (linearized) Gravity Duality invariance and spacetime covariance Conclusions Marc Henneaux Gravitational electric-magnetic duality 8 −1 0 1 2 3 4 5 6 7 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