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Cambridge University Press
Harmonic Superspace

Harmonic Superspace

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An introductory graduate textbook on the harmonic superspace method in extended supersymmetry, that is supersymmetry in which N is two or greater, focusing on the simplest case of N=2. Work on it began when the four authors were at the Bogoliubov Laboratory of Theoretical Physics at JINR, Dubna, where the concept of harmonic superspace was born. Annotation c. Book News, Inc., Portland, OR (

Product Details

ISBN-13: 9780521020428
Publisher: Cambridge University Press
Publication date: 02/22/2007
Series: Cambridge Monographs on Mathematical Physics
Edition description: New Edition
Pages: 324
Product dimensions: 6.61(w) x 9.57(h) x 0.63(d)

Table of Contents

1Introductory overview1
1.1Brief motivations1
1.2Brief summary4
1.3Spaces and superspaces4
1.4Chirality as a kind of Grassmann analyticity6
1.5N = 1 chiral superfields6
1.6Auxiliary fields7
1.7Why standard superspace is not adequate for N = 2 supersymmetry8
1.8Search for conceivable superspaces (spaces)9
1.9N = 2 harmonic superspace10
1.10Dealing with the sphere S[superscript 2]10
1.10.1Comparison with the standard harmonic analysis11
1.11Why harmonic superspace helps13
1.12N = 2 supersymmetric theories15
1.12.1N = 2 matter hypermultiplet15
1.12.2N = 2 Yang-Mills theory16
1.12.3N = 2 supergravity18
1.13N = 3 Yang-Mills theory19
1.14Harmonics and twistors. Self-duality equations20
1.15Chapters of the book and their abstracts23
2Elements of supersymmetry27
2.1Poincare and conformal symmetries27
2.1.1Poincare group27
2.1.2Conformal group28
2.1.3Two-component spinor notation29
2.2Poincare and conformal superalgebras29
2.2.1N = 1 Poincare superalgebra29
2.2.2Extended supersymmetry30
2.2.3Conformal supersymmetry31
2.2.4Central charges from higher dimensions32
2.3Representations of Poincare supersymmetry33
2.3.1Representations of the Poincare group33
2.3.2Poincare superalgebra representations. Massive case34
2.3.3Poincare superalgebra representations. Massless case36
2.3.4Representations with central charge37
2.4Realizations of supersymmetry on fields. Auxiliary fields38
2.4.1N = 1 matter multiplet38
2.4.2N = 1 gauge multiplet41
2.4.3Auxiliary fields and extended supersymmetry41
3.1Coset space generalities44
3.2Coset spaces for the Poincare and super Poincare groups46
3.3N = 2 harmonic superspace50
3.4Harmonic variables54
3.5Harmonic covariant derivatives58
3.6N = 2 superspace with central charge coordinates60
3.7Reality properties61
3.8Harmonics as square roots of quaternions63
4Harmonic analysis66
4.1Harmonic expansion on the two-sphere66
4.2Harmonic integrals67
4.3Differential equations on S[superscript 2]69
4.4Harmonic distributions70
5N = 2 matter with infinite sets of auxiliary fields74
5.1.1N = 1 matter74
5.1.2N = 2 matter multiplets on shell76
5.1.3Relationship between q[superscript +] and w hypermultiplets77
5.1.4Off-shell N = 2 matter before harmonic superspace78
5.2Free off-shell hypermultiplet79
5.2.1The Fayet-Sohnius hypermultiplet constraints as analyticity conditions79
5.2.2Free off-shell q[superscript +] action82
5.2.3Relationship between q[superscript +] and w hypermultiplets off shell85
5.2.4Massive q[superscript +] hypermultiplet86
5.2.5Invariances of the free hypermultiplet actions87
5.3Hypermultiplet self-couplings90
5.3.1General action for q[superscript +] hypermultiplets90
5.3.2An example of a q[superscript +] self-coupling: The Taub-NUT sigma model91
5.3.3Symmetries of the general hypermultiplet action95
5.3.4Analogy with Hamiltonian mechanics98
5.3.5More examples of q[superscript +] self-couplings: The Eguchi-Hanson sigma model and all that99
6N = 2 matter multiplets with a finite number of auxiliary fields. N = 2 duality transformations107
6.1Introductory remarks107
6.2N = 2 tensor multiplet109
6.3The relaxed hypermultiplet111
6.4Further relaxed hypermultiplets112
6.5Non-linear multiplet114
6.6N = 2 duality transformations117
6.6.1Transforming the tensor multiplet118
6.6.2Transforming the relaxed hypermultiplet121
6.6.3Transforming the non-linear multiplet122
6.6.4General criterion for equivalence between hypermultiplet and tensor multiplet actions123
7Supersymmetric Yang-Mills theories128
7.1Gauge fields from matter couplings128
7.1.1N = 0 gauge fields128
7.1.2N = 1 SYM gauge prepotential129
7.1.3N = 2 SYM gauge prepotential131
7.2Superspace differential geometry134
7.2.1General framework135
7.2.2N = 1 SYM theory136
7.2.3N = 2 SYM theory138
7.2.4V[superscript ++] versus Mezincescu's prepotential143
7.3N = 2 SYM action144
8Harmonic supergraphs148
8.1Analytic delta functions148
8.2Green's functions for hypermultiplets150
8.3N = 2 SYM: Gauge fixing, Green's functions and ghosts152
8.4Feynman rules156
8.5Examples of supergraph calculations. Absence of harmonic divergences160
8.6A finite four-point function at two loops167
8.7Ultraviolet finiteness of N = 4, d = 2 supersymmetric sigma models171
9Conformal invariance in N = 2 harmonic superspace175
9.1Harmonic superspace for SU (2, 2|2)175
9.1.1Cosets of SU (2, 2|2)175
9.1.2Structure of the analytic superspace177
9.1.3Transformation properties of the analytic superspace coordinates179
9.1.4Superfield representations of SU (2, 2|2)182
9.2Conformal invariance of the basic N = 2 multiplets185
9.2.2Tensor multiplet186
9.2.3Non-linear multiplet186
9.2.4Relaxed hypermultiplet187
9.2.5Yang-Mills multiplet187
10.1From conformal to Einstein gravity189
10.2N = 1 supergravity192
10.3N = 2 supergravity195
10.3.1N = 2 conformal supergravity: Gauge group and prepotentials195
10.3.2Central charge vielbeins200
10.3.3Covariant harmonic derivative D[superscript --]202
10.3.4Building blocks and superspace densities203
10.3.5Abelian gauge invariance of the Maxwell action206
10.4Different versions of N = 2 supergravity and matter couplings207
10.4.1Principal version of N = 2 supergravity and general matter couplings207
10.4.2Other versions of N = 2 supergravity211
10.5Geometry of N = 2 matter in N = 2 supergravity background214
11Hyper-Kahler geometry in harmonic space217
11.2Preliminaries: Self-dual Yang-Mills equations and Kahler geometry219
11.2.1Harmonic analyticity and SDYM theory220
11.2.2Comparison with the twistor space approach223
11.2.3Complex analyticity and Kahler geometry224
11.2.4Central charge as the origin of the Kahler potential229
11.3Harmonic analyticity and hyper-Kahler potentials232
11.3.1Constraints in harmonic space234
11.3.2Harmonic analyticity236
11.3.3Harmonic derivatives in the [lambda] world238
11.3.4Hyper-Kahler potentials240
11.3.5Gauge choices and normal coordinates244
11.3.6Summary of hyper-Kahler geometry245
11.3.7Central charges as the origin of the hyper-Kahler potentials248
11.3.8An explicit construction of hyper-Kahler metrics253
11.4Geometry of N = 2, d = 4 supersymmetric sigma models256
11.4.1The geometric meaning of the general q[superscript +] action256
11.4.2The component action of the general N = 2 sigma model258
12N = 3 supersymmetric Yang-Mills theory263
12.1N = 3 SYM on-shell constraints263
12.2N = 3 harmonic variables and interpretation of the N = 3 SYM constraints264
12.3Elements of the harmonic analysis on SU (3)/U(1) X U(1)266
12.4N = 3 Grassmann analyticity268
12.5From covariant to manifest analyticity: An equivalent interpretation of the N = 3 SYM constraints271
12.6Off-shell action273
12.7Components on and off shell275
12.8Conformal invariance277
12.9Final remarks279
AppendixNotations, conventions and useful formulas283
A.1Two-component spinors283
A.2Harmonic variables and derivatives284
A.3Spinor derivatives285
A.4Conjugation rules286
A.5Superspace integration measures287

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