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

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Hamiltonian Formalism for Gauge Systems and Manifestly Duality Symmetric Actions Marc Henneaux Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux Introduction Hamiltonian formalism for gauge systems and SO(2)-Electric- magnetic manifestly duality symmetric actions gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in Marc Henneaux D 4 Æ Twisted self-duality Conclusions Quim Fest - September 2015 1 / 20 Quim was one of the first physicists to recognize the beauty and power of the Hamiltonian formalism developed by Dirac for gauge systems, always focusing on physical applications. I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 2 / 20 always focusing on physical applications. I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 2 / 20 I will describe here an application to SO(2) electric-magnetic duality, and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 2 / 20 and larger duality groups. It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 Æ I will describe here an application to SO(2) electric-magnetic SO(2)-Electric- magnetic duality duality, for higher spins in D 4 Æ Twisted self-duality Conclusions 2 / 20 It is motived by the so-called question of “hidden symmetries" of gravitational theories and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 Æ I will describe here an application to SO(2) electric-magnetic SO(2)-Electric- magnetic duality duality, for higher spins in D 4 Æ and larger duality groups. Twisted self-duality Conclusions 2 / 20 and involves both gauge and rigid symmetries. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 Æ I will describe here an application to SO(2) electric-magnetic SO(2)-Electric- magnetic duality duality, for higher spins in D 4 Æ and larger duality groups. Twisted self-duality It is motived by the so-called question of “hidden symmetries" of Conclusions gravitational theories 2 / 20 Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Quim was one of the first physicists to recognize the beauty and Marc Henneaux power of the Hamiltonian formalism developed by Dirac for Introduction gauge systems, SO(2)-Electric- magnetic always focusing on physical applications. gravitational duality in D 4 Æ I will describe here an application to SO(2) electric-magnetic SO(2)-Electric- magnetic duality duality, for higher spins in D 4 Æ and larger duality groups. Twisted self-duality It is motived by the so-called question of “hidden symmetries" of Conclusions gravitational theories and involves both gauge and rigid symmetries. 2 / 20 Following the work of Cremmer and Julia, it has been conjectured 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 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 3 / 20 it has been conjectured 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 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric actions Marc Henneaux Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 3 / 20 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the infinite-dimensional Kac-Moody actions Marc Henneaux algebra E10 Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 3 / 20 8 −1 0 1 2 3 4 5 6 7 or E11 11 10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the infinite-dimensional Kac-Moody actions Marc Henneaux algebra E10 Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 3 / 20 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the infinite-dimensional Kac-Moody actions Marc Henneaux algebra E10 Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in or E D 4 11 Æ Twisted self-duality Conclusions 3 / 20 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the infinite-dimensional Kac-Moody actions Marc Henneaux algebra E10 Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in or E D 4 11 Æ Twisted self-duality Conclusions 3 / 20 8 −1 0 1 2 3 4 5 6 7 11 10 9 8 7 6 5 4 3 2 1 Introduction Hamiltonian formalism for gauge systems and manifestly Following the work of Cremmer and Julia, duality symmetric it has been conjectured that the infinite-dimensional Kac-Moody actions Marc Henneaux algebra E10 Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in or E D 4 11 Æ Twisted self-duality Conclusions might be a “hidden symmetry" of maximal supergravity or of an appropriate extension of it. 3 / 20 One crucial feature of these algebras is that they have duality built in, i.e., they treat democratically all fields and their duals. Whenever a (dynamical) p-form gauge field appears in the spectrum of fields, its dual D p 2-form gauge field also ¡ ¡ appears. Similarly, the graviton and its dual, described by a field with Young symmetry 8 > > > > <> D 3 boxes ¡ > > > > :> simultaneously appear. Introduction Hamiltonian formalism for gauge systems and manifestly duality symmetric actions Marc Henneaux Introduction SO(2)-Electric- magnetic gravitational duality in D 4 Æ SO(2)-Electric- magnetic duality for higher spins in D 4 Æ Twisted self-duality Conclusions 4 / 20 Whenever a (dynamical) p-form gauge field appears in the spectrum of fields, its dual D p 2-form gauge field also ¡ ¡ appears.
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