ebook img

Chern Simons (Super)Gravity PDF

149 Pages·1.5 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Chern Simons (Super)Gravity

100 Years of General Relativity– Vol. 2 CHERN–SIMONS (SUPER)GRAVITY 9863_9789814730938_TP.indd 1 9/12/15 10:16 am 100 Years of General Relativity ISSN: 2424-8223 Series Editor: Abhay Ashtekar (Pennsylvania State University, USA) This series is to publish about two dozen excellent monographs written by top-notch authors from the international gravitational community covering various aspects of the field, ranging from mathematical general relativity through observational ramifications in cosmology, relativistic astrophysics and gravitational waves, to quantum aspects. Published Vol. 1 Numerical Relativity by Masaru Shibata (Kyoto University, Japan) Vol. 2 Chern–Simons (Super)Gravity by Mokhtar Hassaine (Universidad de Talca, Chile) & Jorge Zanelli (Centro de Estudios Científicos, Chile) SongYu - Chern-Simons (Super)Gravity.indd 1 9/12/2015 9:45:18 AM 100 Years of General Relativity– Vol. 2 CHERN–SIMONS (SUPER)GRAVITY Mokhtar Hassaine Universidad de Talca, Chile Jorge Zanelli Centro de Estudios Científicos, Chile World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 9863_9789814730938_TP.indd 2 9/12/15 10:16 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. 100 Years of General Relativity — Vol. 2 CHERN–SIMONS (SUPER)GRAVITY Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4730-93-8 In-house Editor: Song Yu Typeset by Stallion Press Email: [email protected] Printed in Singapore SongYu - Chern-Simons (Super)Gravity.indd 2 9/12/2015 9:45:18 AM December10,2015 6:27 Chern-Simons(Super)Gravity-9.75inx6.5in b2264-fm pagev Preface This book grew out of a set of lecture notes on gravitational Chern-Simons (CS) theories,developedoverthepastdecadeforseveralschoolsanddifferentaudiencesof graduatestudentsandresearchers.Thosenoteswerecirculatedonthe HighEnergy arXiv[1]receivingseveralcomments,correctionsandadditions,untilthelastversion of hep-th/0502193in 2008 [2]. Our interest in CS theories originated in the possibility of constructing gauge- invariant theories that could include gravity consistently. CS gravities are only definedinodddimensionandareaveryspecialclassoftheoriesintheLovelockfam- ily1 that admit local supersymmetric extensions. In those theories, supersymmetry is an off-shell symmetry of the action as in a standard gauge theory [3–5]. From a theoretical point of view, CS forms correspond to a generalized gauge- invariantcouplingbetweenconnections—liketheMaxwellandYang-Millsfields— and charged sources: point-like (particles) or extended (membranes). The simplest example of such interactions is the familiar minimal coupling between the electro- magnetic potential (abelian U(1) connection) and a point charge (0-brane). More generally,a(2n+1)-CSformprovidesanaturalgauge-invariantcouplingbetweena charged2n-braneandanonabelianconnectionone-form.This istrue foranygauge connectionandinparticularforaconnectionoftherelevantalgebrasingravityand supergravity. MostCStheoriesareusefulclassicalorsemiclassicalsystemsandalthoughmany aspects of quantum CS systems have been elucidated, the full quantization of CS field theories in dimensions greater than three is still poorly understood. There is no gauge-invariant perturbative scheme and no natural notion of energy so the usual assumptions of quantum field theory, like the existence of a stable vacuum, 1TheLovelockgravitationtheoriesarethenaturalextensionsofGeneralRelativityfordimensions greaterthanfour(seeChapter4).Theyyieldsecondorderfieldequationsforthemetric,describing thesamedegreesoffreedomasEinstein’stheoryinawaythatisinvariantundergeneralcoordinate transformationsandunderalocalLorentztransformations. v December10,2015 6:27 Chern-Simons(Super)Gravity-9.75inx6.5in b2264-fm pagevi vi Chern-Simons (Super) Gravity are not guaranteed. Nevertheless, apart from the arguments of mathematical ele- gance and beauty, the gravitational CS actions are exceptionally endowed with physical attributes that suggest the viability of a quantum interpretation. CS the- ories are gauge-invariant, scale-invariant and background independent; they have no dimensionful coupling constants,allconstants in the Lagrangianare fixed ratio- nal coefficients that cannot be adjusted without destroying gauge invariance. This exceptional status of CS systems makes them classically interesting to study, and quantum mechanically intriguing yet promising. Acknowledgment It is a pleasure for us to acknowledge a long list of col- laborators, colleagues and students who have taught us a lot and helped us in understanding many subtleties about geometry, Chern-Simons theories, super- symmetry and supergravity in all these years. We have benefited from many enlightening discussions with all of them and some have worked through the manuscript and corrected many misprints and suggested changes to improve the notes. Our thanks go to P. D. Alvarez, L. Alvarez-Gaum´e, A. Anabalo´n, R. Aros, A. Ashtekar, R. Baeza, M. Ban˜ados, G. Barnich, M. Bravo-Gaete, C. Bunster, F. Canfora, L. Castellani, O. Chand´ıa, J. A. de Azca´rraga, N. Deruelle, S. Deser, J. Edelstein, E. Frodden, A. Garbarz, G. Giribet, A. Gomberoff, J. Gomis, M. Henneaux, L. Huerta, R. Jackiw, J. M. F. Labastida, J. Maldacena, C.Mart´ınez,O.Mˇıskovi´c,P.Mora,S.Mukhi,C.Nu´n˜ez,R.Olea,P.Pais,S.Paycha, V.Rivelles,E.Rodr´ıguez,P.Salgado-Arias,P.Salgado-Rebolledo,A.Sen,G.Silva, S.Theisen,F.Toppan,P.Townsend,R.Troncoso,M.ValenzuelaandB.Zwiebach. Thisworkwouldnotexistwithouttheunderstandingandpatientsupportofour families and friends. Our warmthanks for the continued supportand collaboration go to our colleagues and staff at CECs-Valdivia and at University of Talca. This work was supported in part by FONDECYT grants1140155and 1130423. CECSisfundedbytheChileanGovernmentthroughtheCentersofExcellenceBase Funding Programof Conicyt. December10,2015 6:27 Chern-Simons(Super)Gravity-9.75inx6.5in b2264-fm pagevii Contents Preface v 1 The Quantum Gravity Puzzle 1 1.1 Renormalizability and the Triumph of Gauge Theory . . . . . . . . 2 1.1.1 Quantum field theory . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Enter gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Minimal Coupling, Connections and Gauge Symmetry . . . . . . . 5 1.3 Gauge Symmetry and General Coordinate Transformations. . . . . 9 2 Geometry: General Overview 11 2.1 Metric and Affine Structures . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Metric structure . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Nonabelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . 16 3 First Order Gravitation Theory 19 3.1 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 The vielbein . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1.2 The Lorentz connection . . . . . . . . . . . . . . . . . . . . 23 3.1.3 A tale of two covariant derivatives . . . . . . . . . . . . . . 24 3.1.4 Invariant tensors . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Curvature and Torsion Two-Forms . . . . . . . . . . . . . . . . . . 27 3.2.1 Lorentz curvature . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 Riemann and Lorentz curvatures . . . . . . . . . . . . . . . 29 3.2.4 Building blocks. . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Gravity as a Gauge Theory . . . . . . . . . . . . . . . . . . . . . . 31 vii December10,2015 6:27 Chern-Simons(Super)Gravity-9.75inx6.5in b2264-fm pageviii viii Chern-Simons (Super) Gravity 4 Gravity in Higher Dimensions 35 4.1 Lovelock Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Dynamical Content . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 First and second order theories . . . . . . . . . . . . . . . . 40 4.3 Torsional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 Born-Infeld Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Chern-Simons Gravities 47 5.1 Selecting Sensible Theories in Three Dimensions . . . . . . . . . . . 47 5.1.1 Extending the Lorentz group . . . . . . . . . . . . . . . . . 48 5.1.2 Local Poincar´e (quasi-) invariance . . . . . . . . . . . . . . 49 5.1.3 Local (anti-)de Sitter symmetry . . . . . . . . . . . . . . . 50 5.1.4 Three-dimensional zoo. . . . . . . . . . . . . . . . . . . . . 53 5.2 More Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.1 The lesson from D =3 . . . . . . . . . . . . . . . . . . . . 55 5.2.2 Lovelock-Chern-Simons theories . . . . . . . . . . . . . . . 57 5.2.3 Locally Poincar´e invariant theory . . . . . . . . . . . . . . 58 5.3 Torsional Chern-Simons Gravities . . . . . . . . . . . . . . . . . . . 59 6 Additional Features of Chern-Simons Gravity 61 6.1 Lovelock-Chern-Simons Coefficients . . . . . . . . . . . . . . . . . . 61 6.2 Stability of CS-Gravitation Theories . . . . . . . . . . . . . . . . . 62 6.3 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.4 Summary of Gravity Actions . . . . . . . . . . . . . . . . . . . . . . 63 6.5 Finite Action and the Beauty of Gauge Invariance . . . . . . . . . . 63 7 Black Holes, Particles and Branes 67 7.1 Chern-Simons Black Holes . . . . . . . . . . . . . . . . . . . . . . . 68 7.1.1 Three-dimensional black holes . . . . . . . . . . . . . . . . 68 7.1.2 Higher dimensional C-S black holes . . . . . . . . . . . . . 70 7.2 Naked Singularities, Particles and Branes . . . . . . . . . . . . . . . 72 7.2.1 The darker side of the 3D black holes . . . . . . . . . . . . 72 7.2.2 Spinning point particles . . . . . . . . . . . . . . . . . . . . 74 7.2.3 Charged point particles . . . . . . . . . . . . . . . . . . . . 75 7.2.4 (D−3)-branes as conical defects in D dimensions . . . . . 77 7.3 Chern-Simons Branes . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.3.1 The simplest CS branes . . . . . . . . . . . . . . . . . . . . 78 7.3.2 General CS branes . . . . . . . . . . . . . . . . . . . . . . 79 December15,2015 15:48 Chern-Simons(Super)Gravity-9.75inx6.5in b2264-fm pageix Contents ix 8 Supersymmetry and supergravity 81 8.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.1.1 Superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.1.2 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.2 From Rigid Supersymmetry to Supergravity . . . . . . . . . . . . . 85 8.2.1 Standard supergravity . . . . . . . . . . . . . . . . . . . . . 85 8.2.2 AdS superalgebras . . . . . . . . . . . . . . . . . . . . . . . 87 8.3 Fermionic Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3.1 Closing the algebra . . . . . . . . . . . . . . . . . . . . . . 90 9 Chern-Simons Supergravities 93 9.1 AdS CS Supergravity Actions . . . . . . . . . . . . . . . . . . . . . 94 9.1.1 AdS3 supergravity as a CS action in D =3 . . . . . . . . . 95 9.1.2 AdS5 supergravity as a CS action in D =5 . . . . . . . . . 96 9.1.3 Eleven-dimensional AdS CS supergravity action for osp(32|1) . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.2 Poincar´e CS Supergravity Actions . . . . . . . . . . . . . . . . . . . 101 9.2.1 Poincar´e CS supergravity with Majorana spinors . . . . . . 103 10 In¨onu¨-Wigner Contractions and Its Extensions 107 10.1 Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 109 10.1.1 Case without torsion in the action . . . . . . . . . . . . . . 109 10.1.2 Case with torsion in the action . . . . . . . . . . . . . . . . 111 10.2 From osp(32|1) CS Sugra to CS Sugra for the M-Algebra . . . . . . 113 11 Unconventional Supersymmetries 117 11.1 Local Supersymmetry without Gravitini . . . . . . . . . . . . . . . 117 11.2 Matter and Interaction Fields . . . . . . . . . . . . . . . . . . . . . 118 11.3 Combining Matter and Interaction Fields . . . . . . . . . . . . . . . 119 12 Concluding Remarks 123 Bibliography 125 Index 135

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.