Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- 1 Introductory overview -- 1.1 Brief motivations -- 1.2 Brief summary -- 1.3 Spaces and superspaces -- 1.4 Chirality as a kind of Grassmann analyticity -- 1.5 N = 1 chiral superfields -- 1.6 Auxiliary fields -- 1.7 Why standard superspace is not adequate for N = 2 supersymmetry -- 1.8 Search for conceivable superspaces (spaces) -- 1.9 N = 2 harmonic superspace -- 1.10 Dealing with the sphere S -- 1.10.1 Comparison with the standard harmonic analysis -- 1.11 Why harmonic superspace helps -- 1.12 N = 2 supersymmetric theories -- 1.12.1 N = 2 matter hypermultiplet -- 1.12.2 N = 2 Yang-Mills theory -- 1.12.3 N = 2 supergravity -- 1.13 N = 3 Yang-Mills theory -- 1.14 Harmonics and twistors. Self-duality equations -- 1.15 Chapters of the book and their abstracts -- Chapter 1. Introductory overview -- Chapter 2. Elements of supersymmetry -- Chapter 3. Superspace -- Chapter 4. Harmonic analysis -- Chapter 5. N = 2 matter with infinite sets of auxiliary fields -- Chapter 6. N = 2 matter multiplets with a finite number of auxiliary fields. N = 2 duality transformations -- Chapter 7. Supersymmetric Yang-Mills theories -- Chapter 8. Harmonic supergraphs -- Chapter 9. Conformal invariance in N = 2 harmonic superspace -- Chapter 10. Supergravity -- Chapter 11. Hyper-Kähler geometry in harmonic space -- Chapter 12. N = 3 supersymmetric Yang-Mills theory -- 2 Elements of supersymmetry -- 2.1 Poincaré and conformal symmetries -- 2.1.1 Poincaré group -- 2.1.2 Conformal group -- 2.1.3 Two-component spinor notation -- 2.2 Poincaré and conformal superalgebras -- 2.2.1 N = 1 Poincaré superalgebra -- 2.2.2 Extended supersymmetry -- 2.2.3 Conformal supersymmetry -- 2.2.4 Central charges from higher dimensions -- 2.3 Representations of Poincaré supersymmetry.

2.3.1 Representations of the Poincaré group -- 2.3.2 Poincaré superalgebra representations. Massive case -- 2.3.3 Poincaré superalgebra representations. Massless case -- 2.3.4 Representations with central charge -- 2.4 Realizations of supersymmetry on fields. Auxiliary fields -- 2.4.1 N = 1 matter multiplet -- 2.4.2 N = 1 gauge multiplet -- 2.4.3 Auxiliary fields and extended supersymmetry -- 3 Superspace -- 3.1 Coset space generalities -- 3.2 Coset spaces for the Poincaré and super Poincaré groups -- 3.3 N = 2 harmonic superspace -- 3.4 Harmonic variables -- 3.5 Harmonic covariant derivatives -- 3.6 N = 2 superspace with central charge coordinates -- 3.7 Reality properties -- 3.8 Harmonics as square roots of quaternions -- 4 Harmonic analysis -- 4.1 Harmonic expansion on the two-sphere -- 4.2 Harmonic integrals -- 4.3 Differential equations on S -- 4.4 Harmonic distributions -- 5 N = 2 matter with infinite sets of auxiliary fields -- 5.1 Introduction -- 5.1.1 N = 1 matter -- 5.1.2 N = 2 matter multiplets on shell -- 5.1.3 Relationship between q and ω hypermultiplets -- 5.1.4 Off-shell N = 2 matter before harmonic superspace -- 5.2 Free off-shell hypermultiplet -- 5.2.1 The Fayet-Sohnius hypermultiplet constraints as analyticity conditions -- 5.2.2 Free off-shell q action -- 5.2.3 Relationship between q and ω hypermultiplets off shell -- 5.2.4 Massive q hypermultiplet -- 5.2.5 Invariances of the free hypermultiplet actions -- 5.3 Hypermultiplet self-couplings -- 5.3.1 General action for q hypermultiplets -- 5.3.2 An example of a q self-coupling: The Taub-NUT sigma model -- 5.3.3 Symmetries of the general hypermultiplet action -- 5.3.4 Analogy with Hamiltonian mechanics -- 5.3.5 More examples of q self-couplings: The Eguchi-Hanson sigma model and all that.

6 N = 2 matter multiplets with a finite number of auxiliary fields. N = 2 duality transformations -- 6.1 Introductory remarks -- 6.2 N = 2 tensor multiplet -- 6.3 The relaxed hypermultiplet -- 6.4 Further relaxed hypermultiplets -- 6.5 Non-linear multiplet -- 6.6 N = 2 duality transformations -- 6.6.1 Transforming the tensor multiplet -- 6.6.2 Transforming the relaxed hypermultiplet -- 6.6.3 Transforming the non-linear multiplet -- 6.6.4 General criterion for equivalence between hypermultiplet and tensor multiplet actions -- 6.7 Conclusions -- 7 Supersymmetric Yang-Mills theories -- 7.1 Gauge fields from matter couplings -- 7.1.1 N = 0 gauge fields -- 7.1.2 N = 1 SYM gauge prepotential -- 7.1.3 N = 2 SYM gauge prepotential -- 7.2 Superspace differential geometry -- 7.2.1 General framework -- 7.2.2 N = 1 SYM theory -- 7.2.3 N = 2 SYM theory -- 7.2.4 V versus Mezincescu's prepotential -- 7.3 N = 2 SYM action -- 8 Harmonic supergraphs -- 8.1 Analytic delta functions -- 8.2 Green's functions for hypermultiplets -- 8.3 N = 2 SYM: Gauge fixing, Green's functions and ghosts -- 8.4 Feynman rules -- 8.5 Examples of supergraph calculations. Absence of harmonic divergences -- 8.6 A finite four-point function at two loops -- 8.7 Ultraviolet finiteness of N = 4, d = 2 supersymmetric sigma models -- 9 Conformal invariance in N = 2 harmonic superspace -- 9.1 Harmonic superspace for SU(2, 2

10.2 N = 1 supergravity -- 10.3 N = 2 supergravity -- 10.3.1 N = 2 conformal supergravity: Gauge group and prepotentials -- 10.3.2 Central charge vielbeins -- 10.3.3 Covariant harmonic derivative D -- 10.3.4 Building blocks and superspace densities -- 10.3.5 Abelian gauge invariance of the Maxwell action -- 10.4 Different versions of N = 2 supergravity and matter couplings -- 10.4.1 Principal version of N = 2 supergravity and general matter couplings -- 10.4.2 Other versions of N = 2 supergravity -- 10.5 Geometry of N = 2 matter in N = 2 supergravity background -- 11 Hyper-Kähler geometry in harmonic space -- 11.1 Introduction -- 11.2 Preliminaries: Self-dual Yang-Mills equations and Kähler geometry -- 11.2.1 Harmonic analyticity and SDYM theory -- 11.2.2 Comparison with the twistor space approach -- 11.2.3 Complex analyticity and Kähler geometry -- 11.2.4 Central charge as the origin of the Kähler potential -- 11.3 Harmonic analyticity and hyper-Kähler potentials -- 11.3.1 Constraints in harmonic space -- 11.3.2 Harmonic analyticity -- 11.3.3 Harmonic derivatives in the λ world -- 11.3.4 Hyper-Kähler potentials -- 11.3.5 Gauge choices and normal coordinates -- 11.3.6 Summary of hyper-Kähler geometry -- 11.3.7 Central charges as the origin of the hyper-Kähler potentials -- 11.3.8 An explicit construction of hyper-Kähler metrics -- 11.4 Geometry of N =2, d = 4 supersymmetric sigma models -- 11.4.1 The geometric meaning of the general q action -- 11.4.2 The component action of the general N =2 sigma model -- 12 N = 3 supersymmetric Yang-Mills theory -- 12.1 N = 3 SYM on-shell constraints -- 12.2 N = 3 harmonic variables and interpretation of the N = 3 SYM constraints -- 12.3 Elements of the harmonic analysis on SU(3)/U(1) × U(1) -- 12.4 N = 3 Grassmann analyticity.

12.5 From covariant to manifest analyticity: An equivalent interpretation of the N = 3 SYM constraints -- 12.6 Off-shell action -- 12.7 Components on and off shell -- 12.8 Conformal invariance -- 12.9 Final remarks -- 13 Conclusions -- Appendix Notations, conventions and useful formulas -- A.1 Two-component spinors -- A.2 Harmonic variables and derivatives -- A.3 Spinor derivatives -- A.4 Conjugation rules -- A.5 Superspace integration measures -- References -- Index.