HARMONIC SUPERSPACE

By M. NOAND

CONTENTS INTRODUCTORY OVERVIEW BRIEF MOTIVATIONS BRIEF SUMMARY SPACES AND SUPERSPACES CHIRALITY AS A KIND OF GRASSMANN ANALYTICITY N = 1 CHIRAL SUPER?ELDS AUXILIARY ?ELDS WHY STANDARD SUPERSPACE IS NOT ADEQUATE FOR N = 2 SUPERSYMMETRY SEARCH FOR CONCEIVABLE SUPERSPACES (SPACES) N = 2 HARMONIC SUPERSPACE 1101 COMPARISON WITH THE STANDARD HARMONIC ANALYSIS 111 WHY HARMONIC SUPERSPACE HELPS N = 2 SUPERSYMMETRIC THEORIES 1121 N = 2 MATTER HYPERMULTIPLET 1122 N = 2 YANG–MILLS THEORY 1123 N = 2 SUPERGRAVITY N = 3 YANG–MILLS THEORY 114 HARMONICS AND TWISTORS SELF-DUALITY EQUATIONS 115 CHAPTERS OF THE BOOK AND THEIR ABSTRACTS ELEMENTS OF SUPERSYMMETRY POINCAR´E AND CONFORMAL SYMMETRIES 211 POINCAR´E GROUP 212 CONFORMAL GROUP 213 TWO-COMPONENT SPINOR NOTATION POINCAR´E AND CONFORMAL SUPERALGEBRAS 221 N = 1 POINCAR´E SUPERALGEBRA 222 EXTENDED SUPERSYMMETRY 223 CONFORMAL SUPERSYMMETRY 224 CENTRAL CHARGES FROM HIGHER DIMENSIONS REPRESENTATIONS OF POINCAR´E SUPERSYMMETRY 231 REPRESENTATIONS OF THE POINCAR´E GROUP 232 POINCAR´E SUPERALGEBRA REPRESENTATIONS MASSIVE CASE 233 POINCAR´E SUPERALGEBRA REPRESENTATIONS MASSLESS CASE 234 REPRESENTATIONS WITH CENTRAL CHARGE REALIZATIONS OF SUPERSYMMETRY ON ?ELDS AUXILIARY ?ELDS 241 N = 1 MATTER MULTIPLET 242 N = 1 GAUGE MULTIPLET 243 AUXILIARY ?ELDS AND EXTENDED SUPERSYMMETRY SUPERSPACE COSET SPACE GENERALITIES COSET SPACES FOR THE POINCAR´E AND SUPER POINCAR´E GROUPS HARMONIC VARIABLES HARMONIC COVARIANT DERIVATIVES N = 2 SUPERSPACE WITH CENTRAL CHARGE COORDINATES REALITY PROPERTIES HARMONICS AS SQUARE ROOTS OF QUATERNIONS HARMONIC ANALYSIS HARMONIC EXPANSION ON THE TWO-SPHERE HARMONIC INTEGRALS HARMONIC DISTRIBUTIONS N = 2 MATTER WITH IN?NITE SETS OF AUXILIARY ?ELDS INTRODUCTION 511 N = 1 MATTER 512 N = 2 MATTER MULTIPLETS ON SHELL 513 RELATIONSHIP BETWEEN Q+ AND ? HYPERMULTIPLETS 514 OFF-SHELL N = 2 MATTER BEFORE HARMONIC SUPERSPACE FREE OFF-SHELL HYPERMULTIPLET 521 THE FAYET–SOHNIUS HYPERMULTIPLET CONSTRAINTS AS ANALYTICITY CONDITIONS 522 FREE OFF-SHELL Q+ ACTION 523 RELATIONSHIP BETWEEN Q+ AND ? HYPERMULTIPLETS OFF SHELL 524 MASSIVE Q+ HYPERMULTIPLET 525 INVARIANCES OF THE FREE HYPERMULTIPLET ACTIONS HYPERMULTIPLET SELF-COUPLINGS 531 GENERAL ACTION FOR Q+ HYPERMULTIPLETS 532 AN EXAMPLE OF A Q+ SELF-COUPLING: THE TAUB–NUT SIGMA MODEL 533 SYMMETRIES OF THE GENERAL HYPERMULTIPLET ACTION 534 ANALOGY WITH HAMILTONIAN MECHANICS 535 MORE EXAMPLES OF Q+ SELF-COUPLINGS: THE EGUCHI–HANSON SIGMA MODEL AND ALL THAT N = 2 MATTER MULTIPLETS WITH A ?NITE NUMBER OF AUXILIARY ?ELDS N = 2 DUALITY TRANSFORMATIONS INTRODUCTORY REMARKS N = 2 TENSOR MULTIPLET THE RELAXED HYPERMULTIPLET FURTHER RELAXED HYPERMULTIPLETS NON-LINEAR MULTIPLET N = 2 DUALITY TRANSFORMATIONS 661 TRANSFORMING THE TENSOR MULTIPLET 662 TRANSFORMING THE RELAXED HYPERMULTIPLET 663 TRANSFORMING THE NON-LINEAR MULTIPLET 664 GENERAL CRITERION FOR EQUIVALENCE BETWEEN HYPERMULTIPLET AND TENSOR MULTIPLET ACTIONS CONCLUSIONS SUPERSYMMETRIC YANG–MILLS THEORIES GAUGE ?ELDS FROM MATTER COUPLINGS 711 N = 0 GAUGE ?ELDS 712 N = 1 SYM GAUGE PREPOTENTIAL 713 N = 2 SYM GAUGE PREPOTENTIAL SUPERSPACE DIFFERENTIAL GEOMETRY 721 GENERAL FRAMEWORK 722 N = 1 SYM THEORY 723 N = 2 SYM THEORY 724 V ++ VERSUS MEZINCESCU’S PREPOTENTIAL N = 2 SYM ACTION HARMONIC SUPERGRAPHS ANALYTIC DELTA FUNCTIONS GREEN’S FUNCTIONS FOR HYPERMULTIPLETS N = 2 SYM: GAUGE ?XING, GREEN’S FUNCTIONS AND GHOSTS FEYNMAN RULES EXAMPLES OF SUPERGRAPH CALCULATIONS ABSENCE OF HARMONIC DIVERGENCES A ?NITE FOUR-POINT FUNCTION AT TWO LOOPS ULTRAVIOLET ?NITENESS OF N = 4, D = 2 SUPERSYMMETRIC SIGMA MODELS CONFORMAL INVARIANCE IN N = 2 HARMONIC SUPERSPACE HARMONIC SUPERSPACE FOR SU(2, 2|2) 911 COSETS OF SU(2, 2|2) 912 STRUCTURE OF THE ANALYTIC SUPERSPACE 913 TRANSFORMATION PROPERTIES OF THE ANALYTIC SUPERSPACE COORDINATES 914 SUPER?ELD REPRESENTATIONS OF SU(2, 2|2) CONFORMAL INVARIANCE OF THE BASIC N = 2 MULTIPLETS 921 HYPERMULTIPLET 922 TENSOR MULTIPLET 923 NON-LINEAR MULTIPLET 924 RELAXED HYPERMULTIPLET 925 YANG–MILLS MULTIPLET SUPERGRAVITY 101 FROM CONFORMAL TO EINSTEIN GRAVITY N = 1 SUPERGRAVITY N = 2 SUPERGRAVITY 1031 N = 2 CONFORMAL SUPERGRAVITY: GAUGE GROUP AND PREPOTENTIALS 1032 CENTRAL CHARGE VIELBEINS 1033 COVARIANT HARMONIC DERIVATIVE D?? 1034 BUILDING BLOCKS AND SUPERSPACE DENSITIES 1035 ABELIAN GAUGE INVARIANCE OF THE MAXWELL ACTION 104 DIFFERENT VERSIONS OF N = 2 SUPERGRAVITY AND MATTER COUPLINGS 1041 PRINCIPAL VERSION OF N = 2 SUPERGRAVITY AND GENERAL MATTER COUPLINGS 1042 OTHER VERSIONS OF N = 2 SUPERGRAVITY 105 GEOMETRY OF N = 2 MATTER IN N = 2 SUPERGRAVITY BACKGROUND HYPER-K¨AHLER GEOMETRY IN HARMONIC SPACE 111 INTRODUCTION PRELIMINARIES: SELF-DUAL YANG–MILLS EQUATIONS AND K¨AHLER GEOMETRY 1121 HARMONIC ANALYTICITY AND SDYM THEORY 1122 COMPARISON WITH THE TWISTOR SPACE APPROACH 1123 COMPLEX ANALYTICITY AND K¨AHLER GEOMETRY 1124 CENTRAL CHARGE AS THE ORIGIN OF THE K¨AHLER POTENTIAL 113 HARMONIC ANALYTICITY AND HYPER-K¨AHLER POTENTIALS 1131 CONSTRAINTS IN HARMONIC SPACE 1132 HARMONIC ANALYTICITY 1133 HARMONIC DERIVATIVES IN THE ? WORLD 1134 HYPER-K¨AHLER POTENTIALS 1135 GAUGE CHOICES AND NORMAL COORDINATES 1136 SUMMARY OF HYPER-K¨AHLER GEOMETRY 1137 CENTRAL CHARGES AS THE ORIGIN OF THE HYPER-K¨AHLER POTENTIALS 1138 AN EXPLICIT CONSTRUCTION OF HYPER-K¨AHLER METRICS 114 GEOMETRY OF N = 2, D = 4 SUPERSYMMETRIC SIGMA MODELS 1141 THE GEOMETRIC MEANING OF THE GENERAL Q+ ACTION 1142 THE COMPONENT ACTION OF THE GENERAL N = 2 SIGMA MODEL N = 3 SUPERSYMMETRIC YANG–MILLS THEORY N = 3 SYM ON-SHELL CONSTRAINTS N = 3 HARMONIC VARIABLES AND INTERPRETATION OF THE N = 3 SYM CONSTRAINTS 123 ELEMENTS OF THE HARMONIC ANALYSIS ON SU(3)/U(1) × U(1) N = 3 GRASSMANN ANALYTICITY FROM COVARIANT TO MANIFEST ANALYTICITY: AN EQUIVALENT INTERPRETATION OF THE N = 3 SYM CONSTRAINTS 126 OFF-SHELL ACTION 127 COMPONENTS ON AND OFF SHELL 128 CONFORMAL INVARIANCE 129 FINAL REMARKS REFERENCES