Table Of ContentSPRINGER BRIEFS IN MATHEMATICAL PHYSICS 1
Yoshiaki Tanii
Introduction
to Supergravity
SpringerBriefs in Mathematical Physics
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Yoshiaki Tanii
Introduction to Supergravity
123
YoshiakiTanii
Graduate School ofScience
andEngineering
Saitama University
Saitama
Japan
ISSN 2197-1757 ISSN 2197-1765 (electronic)
ISBN 978-4-431-54827-0 ISBN 978-4-431-54828-7 (eBook)
DOI 10.1007/978-4-431-54828-7
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Preface
This book is a pedagogical introduction to supergravity. Supergravity is a gravi-
tational field theory that includes supersymmetry (symmetry between bosons and
fermions) and is a generalization of Einstein’s general relativity. Supergravity
provides a low energy effective theory of superstring theory, which has attracted
muchattentionasacandidatefortheunifiedtheoryoffundamentalparticles,andit
is a useful tool for studying nonperturbative properties of superstring theory such
as D-branes and string duality.
This work considers classical supergravities in four and higher dimensional
spacetimewiththeirapplicationstosuperstringtheoryinmind.Moreconcretely,it
discusses classical Lagrangians (or field equations) and symmetry properties of
supergravities. Besides local symmetries, supergravities often have global non-
compact symmetries, which play a crucial role intheir applicationsto superstring
theory. One of the main features of this book is its detailed discussions of these
non-compact symmetries.
Theaimofthebookistwofold.Oneistoexplainthebasicideasofsupergravity
to those who are not familiar with it. Toward that end, the discussions are made
both pedagogical and concrete by stating equations explicitly. The other is to
collect relevant formulae in one place so as to be useful for applications to string
theory.Theyincludethelistsofpossibletypesofspinorsineachdimension,field
contents of supergravities, and global symmetries of supergravities.
Most of the discussions are restricted to pure supergravities without matter
couplings. An exception is a coupling to the super Yang–Mills multiplet in ten
dimensions. Supergravities in lower than four dimensions are not considered.
Therearemanyotherissuesonsupergravitywhicharenotdiscussedinthisbook.
For those issues, consult the references given at the end of Sect. 1.1.
The plan of the book is as follows. In Chap. 1, we first explain a role of
supergravityinsuperstringtheorybriefly,andthenreviewtheformulationsofthe
gravitational field and other fields coupled to it. In Chap. 2, we discuss super-
gravitiesinfourdimensionsindetails.Muchofthepropertiesofsupergravitiesin
higher dimensions already appear in four dimensions. In Chap. 3, we discuss
superalgebras and supermultiplets in general dimensions and give the lists of
possible types of supergravities and their field contents. In Chap. 4, we consider
global non-compact symmetries in supergravities, which are useful in under-
standing the structure of scalar fields. The non-compact symmetry in
v
vi Preface
supergravities is sometimes realized as a duality symmetry of vector or antisym-
metric tensor fields, which is a generalizations of the electric–magnetic duality in
Maxwell’sequations.InChap.5,weconsidersupergravitiesinhigherdimensions.
In particular, the Lagrangians (or field equations) and the symmetry properties of
supergravities in 11 and 10 dimensions are discussed in details. In Chap. 6, we
considerdimensionalreductionsofsupergravitiesinelevenandtendimensionsto
lower dimensions in order to understand the origins of the global non-compact
symmetries. Finally, in Chap. 7, we consider gauged supergravities, which have
minimal couplings to vector gauge fields, and massive supergravities similar to
them.NotationandconventionsusedinthisbookaresummarizedinAppendixA.
Formulae of gamma matrices and spinors in general dimensions are collected in
Appendix B.
Saitama, April 2014 Yoshiaki Tanii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Supergravity and Superstring . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Gravitational Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Metric Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Vielbein Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Yang–Mills Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Antisymmetric Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Dual Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Self-dual Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.3 Massive Chern–Simons Type Theory . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Supergravities in Four Dimensions. . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Superalgebras and Supermultiplets. . . . . . . . . . . . . . . . . . . . . . 17
2.2 Supersymmetric Field Theories. . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 N¼1 Poincaré Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Local Supersymmetry of N¼1 Poincaré Supergravity. . . . . . . 24
2.4.1 Invariance of the Action . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 Commutator Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 N¼1 Anti de Sitter Supergravity . . . . . . . . . . . . . . . . . . . . . 26
2.6 Extended Supersymmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 N¼2 Poincaré Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8 N¼2 Anti de Sitter Supergravity . . . . . . . . . . . . . . . . . . . . . 32
2.9 N(cid:2)3 Supergravities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Superalgebras and Supermultiplets. . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Spinors in General Dimensions. . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Gamma Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2 Dirac Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3 Weyl Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.4 Majorana Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.5 Majorana–Weyl Spinors. . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.6 Symplectic Majorana Spinors. . . . . . . . . . . . . . . . . . . . 43
vii
viii Contents
3.2 Super Poincaré Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Supermultiplets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Massless Sectors of M Theory and Superstring Theory . . . . . . . 47
3.5 Super Anti de Sitter Algebras. . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Global Non-compact Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Non-linear Sigma Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 SLð2; RÞ=SOð2Þ Non-linear Sigma Model . . . . . . . . . . . 55
4.2 Duality Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Duality Symmetry in General Even Dimensions. . . . . . . 58
4.2.2 Compact Duality Symmetry. . . . . . . . . . . . . . . . . . . . . 61
4.2.3 Non-compact Duality Symmetry. . . . . . . . . . . . . . . . . . 62
4.3 D¼4, N¼8 Poincaré Supergravity . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Poincaré Supergravities in Higher Dimensions . . . . . . . . . . . . . . . 71
5.1 General Structure of Poincaré Supergravities . . . . . . . . . . . . . . 71
5.2 D¼11, N¼1 Poincaré Supergravity . . . . . . . . . . . . . . . . . . 72
5.3 D¼10, N¼ð1;1Þ Poincaré Supergravity. . . . . . . . . . . . . . . . 73
5.4 D¼10, N¼ð2;0Þ Poincaré Supergravity. . . . . . . . . . . . . . . . 76
5.5 D¼10, N¼ð1;0Þ Poincaré Supergravity. . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 Dimensional Reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Compactifications and Dimensional Reductions . . . . . . . . . . . . 89
6.2 Dimensional Reductions of Field Theories . . . . . . . . . . . . . . . . 91
6.2.1 Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2.2 Yang–Mills Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.3 Antisymmetric Tensor Field. . . . . . . . . . . . . . . . . . . . . 96
6.3 Dimensional Reductions of D¼11, N¼1 Supergravity . . . . . 98
6.3.1 D¼10 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.2 D¼9 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.3 D¼8 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.4 D¼7 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.5 D¼6 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3.6 D¼5 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.7 D¼4 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Dimensional Reductions of D¼10, N¼ð2;0Þ
Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Dimensional Reductions of D¼10, N¼ð1;0Þ
Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents ix
7 Gauged Supergravities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1 Gauged Supergravities and Massive Supergravities . . . . . . . . . . 111
7.2 D¼4, N¼8 Gauged Supergravity. . . . . . . . . . . . . . . . . . . . 112
7.3 Gauged Supergravities in Higher Dimensions. . . . . . . . . . . . . . 116
7.3.1 D¼7, N¼4 Gauged Supergravity. . . . . . . . . . . . . . . 116
7.3.2 D¼5, N¼8 Gauged Supergravity. . . . . . . . . . . . . . . 117
7.4 D¼10, N¼ð1;1Þ Massive Supergravity. . . . . . . . . . . . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Appendix A: Notation and Conventions. . . . . . . . . . . . . . . . . . . . . . . 121
Appendix B: Formulae of Gamma Matrices. . . . . . . . . . . . . . . . . . . . 123
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Chapter 1
Introduction
1.1 SupergravityandSuperstring
Supergravityisatheoryofgravitywhichhassupersymmetry,asymmetrybetween
bosons and fermions. Supersymmetry in supergravity is a local symmetry like the
gauge symmetry in the standard theory of particle physics. The gauge field of the
localsupersymmetryistheRarita–Schwingerfield,whichrepresentsaparticlewith
spin 3 calledagravitino.Supergravityalsohasalocalsymmetryunderthegeneral
2
coordinatetransformation,whosegaugefieldisthegravitationalfield.
Animportantroleofsupergravityisinitsrelationtosuperstringtheory.Super-
stringtheoryisacandidateoftheunifiedtheoryoffundamentalparticlesincluding
gravity.Itisexpectedthatsuperstringtheoryprovidesaconsistentquantumtheoryof
gravitysinceithasnoultravioletdivergence.Superstringtheorycontainsmassless
states such as graviton and gauge particles in addition to infinitely many massive
states. The mass scale of the theory is supposed to be very large, say the Planck
energyscale,andonlythemasslessstatesareimportantatlowenergy.Supergravity
providesalowenergyeffectivetheoryofthemasslesssectorofsuperstringtheory
andcanbeusedtostudyitslowenergyproperties.
Superstring theory is an incomplete theory at present as its fundamental
formulationisnotknown.Asaconsequence,itisdifficulttostudyitsnon-perturbative
properties,whoseunderstandingisindispensableforapplicationstophysics.Inthe
1990s there was considerable progress in non-perturbative understanding of string
theory, which was possible by two important discoveries: D-branes [8] and string
duality[4, 16].
D-branesarespatiallyextendedobjectsappearinginstringtheoryandcorrespond
tosolitonsinfieldtheories.Assolitonsareusefultoolsinnon-perturbativestudyof
field theories, D-branes can be used to study non-perturbative properties of string
theory.Atlowenergy,D-branesarerepresentedbyclassicalsolutionsoftheeffective
supergravitytheory.OnecanstudyvariousaspectsofD-branesbyusingsupergravity.
Stringdualityistherelationamongseeminglydifferentstringtheories.Fivesuper-
symmetricstringtheoriesareknown.TheyaretypeI,typeIIA,typeIIBsuperstring
Y.Tanii,IntroductiontoSupergravity,SpringerBriefsinMathematicalPhysics, 1
DOI:10.1007/978-4-431-54828-7_1,©TheAuthor(s)2014
