Supersymmetric Mechanics - Vol. 1: Supersymmetry, Noncommutativity and Matrix Models (Lecture Notes in Physics)

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Preface

This is the third volume in a series of books on the general topics of Supersymmetric Mechanics, with the first and second volumes being published as Lecture Notes in Physics Vol. 698, Supersymmetric Mechanics – Vol. 1: Supersymmetry, Noncommutativity and Matrix Models (ISBN: 3-540-33313-4), and Lecture Notes in Physics Vol. 701, Supersymmetric Mechanics – Vol. 2: The Attractor Mechanism and Space Time Singularities (ISBN: 3-540-34156-0). The aim of this ongoing collection is to provide a reference corpus of suitable, introductory material to the field, by gathering the significantly expanded and edited versions of all tutorial lectures, given over the years at the well-established annual INFN-Laboratori Nazionali di Frascati Winter School on the Attractor Mechanism, directed by myself. The present set of notes results again from the participation and dedication of prestigious lecturers, such as Iosif Bena, Sergio Ferrara, Renata Kallosh, Per Kraus, Finn Larsen, and Boris Pioline. As usual, the lectures were subsequently carefully edited and reworked, taking into account the extensive follow-up discussions. The present volume emphasizes topics of great recent interest, namely general concepts of attractors in supersymmetric gravity and black holes. A two-parameter family of spherically symmetric, static, asymptotically flat, electrically charged singular metrics in d = 4 is described by the so-called Reissner-N\00 ordstrom solution. It may be rigorously shown that the spherically symmetric solution of N = 2, d = 4 Maxwell-Einstein supergravity represented by an extremal Reissner-N\00 ordstrom black hole preserves one-half of the supersymmetry isometries out of the eight related to the asymptotical limit given by the N = 2, d = 4 Minkowski background. When approaching the event horizon of the black hole, one gets a restoration of the previously lost four additional supersymmetries, hence reobtaining a maximally symmetric N = 2 metric background, namely the 4-d BertottiRobinson AdS × S2 black hole metric. In the earlier book Supersymmetric Mechanics – Vol. 2, a general dynamical principle was considered, namely the “attractor mechanism”, which

FM.pdf

This is the new Preface – CE is necessary

-------------------------------------------------------------------------------This is the third volume in a series of books on the general topics of Supersymmetric Mechanics, with the first and second volumes being published as Lecture Notes in Physics Vol. 698, Supersymmetric Mechanics - Vol. 1: Supersymmetry, Noncommutativity and Matrix Models (ISBN: 3-540-33313-4), and Lecture Notes in Physics Vol. 701, Supersymmetric Mechanics - Vol. 2: The Attractor Mechanism and Space Time Singularities (ISBN: 3-540-34156-0). The aim of this ongoing collection is to provide a reference corpus of suitable, introductory material to the field, by gathering the significantly expanded and edited versions of all tutorial lectures, given over the years at the well established annual INFN-Laboratori Nazionali di Frascati Winter School on the Attractor Mechanism, directed by myself. The present set of notes results again from the participation and dedication of prestigious lecturers, such as Iosif Bena, Sergio Ferrara, Renata Kallosh, Per Kraus, Finn Larsen and Boris Pioline. As usual, the lectures were subsequently carefully edited and reworked, taking into account the extensive follow-up discussions. The present volume emphasizes topics of great recent interest, namely general concepts of attractors in supersymmetric gravity and black holes. A two-parameter family of spherically symmetric, static, asymptotically flat, electrically charged singular metrics in $d =4$ is described by the so-called Reissner-N\"ordstrom solution. It may be rigorously shown that the spherically symmetric solution of $N=2, d=4$ Maxwell-Einstein supergravity represented by an extremal Reissner-N\"ordstrom black hole preserves one-half of the supersymmetry isometries out of the eight related to the asymptotical limit given by the $N=2, d=4$ Minkowski background. When approaching the event horizon of the black hole, one gets a restoration of the previously lost four additional supersymmetries, hence reobtaining a maximally symmetric $N =2$ metric background, namely the 4-d Bertotti-Robinson $AdS\times S^2$ black hole metric. In the earlier book Supersymmetric Mechanics - Vol. 2 a general dynamical principle was considered, namely the ``attractor mechanism'', which governs the dynamics inside the moduli space, with supersymmetry being related to dynamical systems with fixed points describing the corresponding equilibrium state and the stability properties. If this mechanism holds, in approaching some fixed values, which depend solely upon the electric and magnetic charges of the theory, the orbits of the dynamical evolution lose all memory of their initial conditions, and yet the overall dynamics remains fully deterministic. Historically, the first attractor example in supersymmetric systems emerged from the consideration of extreme black holes in $N =2, d=4,5$ MaxwellEinstein supergravities coupled with matter multiplets. In the present volume, some of the founders of the research in this field, interacting among themselves, as well as with younger collaborators, yield a pedagogical introduction to the subject. In his lectures Iosif Bena (co-authored by Nick Warner) gives an introduction to the construction and analysis of three-charge configurations in string theory and supergravity and described the corresponding implications for the physics of black holes in string theory. Sergio Ferrara (co-authored by Mike Duff) reviews some recently established connections between the mathematics of black hole entropy in string theory and that of multipartite entanglement in quantum information theory, a topic that could be of great interest also for experimental testing and perhaps even for potential applications. The lectures by Renata Kallosh (co-authored by Stefano Bellucci, Sergio Ferrara and Alessio Marrani), provides a pedagogical, introductory review of the Attractor Mechanism (at work in two different $4$-dimensional frameworks: extremal black holes in $N=2$ supergravity and $N=1$ flux compactifications. AdS$_3$ black holes and their connection to two-dimensional conformal field theories via the AdS/CFT correspondence are the subject of the lectures by Per Kraus, including background material on gravity in AdS$_3$, in the context of the holographic renormalization. Also Finn Larsen in his lectures yields a pedagogical introduction to the attractor mechanism, in particular in five dimensions, concentrating chiefly on supersymmetry-preserving black holes in five dimensions, both with and without spherical symmetry, being motivated essentially by the consideration of black rings, as well as rotating black holes. Pioline in his contribution "Black Holes, Topological Strings and Quantum Attractors" reviews recent developments on the relation between the macroscopic entropy of fourdimensional BPS black holes and the microscopic counting of states. I wish to thank all lecturers and participants of the School for contributing to create an almost magical

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governs the dynamics inside the moduli space, with supersymmetry being related to dynamical systems with fixed points describing the corresponding equilibrium state and the stability properties. If this mechanism holds, in approaching some fixed values, which depend solely upon the electric and magnetic charges of the theory, the orbits of the dynamical evolution lose all memory of their initial conditions, and yet the overall dynamics remains fully deterministic. Historically, the first attractor example in supersymmetric systems emerged from the consideration of extreme black holes in N = 2, d = 4, 5 Maxwell-Einstein supergravities coupled with matter multiplets. In the present volume, some of the founders of the research in this field, interacting among themselves, as well as with younger collaborators, yield a pedagogical introduction to the subject. In his lectures, Iosif Bena (co-authored by Nick Warner) gives an introduction to the construction and analysis of three-charge configurations in string theory and supergravity and describes the corresponding implications for the physics of black holes in string theory. Sergio Ferrara (co-authored by Mike Duff) reviews some recently established connections between the mathematics of black hole entropy in string theory and that of multipartite entanglement in quantum information theory, a topic that could be of great interest also for experimental testing and perhaps even for potential applications. The lectures by Renata Kallosh (co-authored by Stefano Bellucci, Sergio Ferrara, and Alessio Marrani) provides a pedagogical, introductory review of the Attractor Mechanism (at work in two different 4-dimensional frameworks: extremal black holes in N = 2 supergravity and N = 1 flux compactifications. AdS3 black holes and their connection to two-dimensional conformal field theories via the AdS/CFT correspondence are the subject of the lectures by Per Kraus, including background material on gravity in AdS3 , in the context of the holographic renormalization. Also Finn Larsen in his lectures yields a pedagogical introduction to the attractor mechanism, in particular in five dimensions, concentrating chiefly on supersymmetry-preserving black holes in five dimensions, both with and without spherical symmetry, being motivated essentially by the consideration of black rings, as well as rotating black holes. Pioline in his contribution “Black Holes, Topological Strings and Quantum Attractors” reviews recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states. I wish to thank all lecturers and participants of the School for contributing to create an almost magical atmosphere to progress in the learning and the further researching in this absolutely fascinating topic. I wish to thank most warmly Mrs. Silvia Colasanti for ¡??¿ generous efforts in the secretarial work and in various organizational aspects. My gratitude goes to INFN and in particular to Mario Calvetti for supporting the School. In welcoming our brand new daughter Erica, my thoughts go to my wife Gloria and our beloved Costanza, Eleonora, and Annalisa for providing me everyday joy, without which I could never have accomplished this effort. Frascati, December 2007

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atmosphere to progress in the learning and the further researching in this absolutely fascinating topic. I wish to thank most warmly Mrs. Silvia Colasanti for generous efforts in the secretarial work and in various organizational aspects. My gratitude goes to INFN and in particular to Mario Calvetti for supporting the School. In welcoming our brand new daughter Erica, my thoughts go to my wife Gloria and our beloved Costanza, Eleonora and Annalisa for providing me everyday joy, without which I could never have accomplished this effort. Frascati, December 2007 Stefano Bellucci

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Stefano Bellucci