Chern Simons (Super)Gravity

149 Pages·1.5 MB·English

by  Mokhtar Hassaine, Jorge Zanelli

100 Years of General Relativity– Vol. 2
CHERN–SIMONS
(SUPER)GRAVITY
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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)
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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
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100 Years of General Relativity — Vol. 2
CHERN–SIMONS (SUPER)GRAVITY
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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
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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.
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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
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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
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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