Supersymmetric Mechanics - Vol. 1 Book

1 A Journey Through Garden Algebras S. Bellucci S.J. Gates E. Orazi 1

1.1 Introduction 1

1.2 GR(d, N) Algebras 3

1.2.1 Geometrical Interpretation of GR(d, N) Algebras 3

1.2.2 Twisted Representations 6

1.2.3 GR(d, N) Algebras Representation Theory 7

1.3 Relationships Between Different Models 13

1.3.1 Automorphic Duality Transformations 13

1.3.2 Reduction 17

1.4 Applications 18

1.4.1 Spinning Particle 18

1.4.2 N = 8 Unusual Representations 27

1.5 Graphical Supersymmetric Representation Technique: Adinkras 28

1.5.1 N = 1 Supermultiplets 29

1.5.2 N = 2 Supermultiplets 31

1.5.3 Adinkras Folding 33

1.5.4 Escheric Supermultiplets 34

1.5.5 Through Higher N 37

1.5.6 Gauge Invariance 43

1.6 Conclusions 44

References 46

2 Supersymmetric Mechanics in Superspace S. Bellucci S. Krivonos 49

2.1 Introduction 49

2.2 Supersymmetry in d = 1 49

2.2.1 Super-Poincare´ Algebra in d = 1 50

2.2.2 Auxiliary Fields 52

2.2.3 Superfields 54

2.2.4 N = 4 Supermultiplets 56

2.3 Nonlinear Realizations 63

2.3.1 Realizations in the Coset Space 63

2.3.2 Realizations: Examples and Technique 65

2.3.3 Cartan's Forms 67

2.3.4 Nonlinear Realizations and Supersymmetry 71

2.4 N = 8 Supersymmetry 76

2.4.1 N = 8, d = 1 Superspace 77

2.4.2 N = 8, d = 1 Supermultiplets 78

2.4.3 Supermultiplet (4, 8, 4) 85

2.4.4 Supermultiplet (5, 8, 3) 88

2.4.5 Supermultiplet (6, 8, 2) 90

2.4.6 Supermultiplet (7, 8, 1) 92

2.4.7 Supermultiplet (8, 8, 0) 92

2.5 Summary and Conclusions 94

References 94

3 Noncommutative Mechanics, Landau Levels, Twistors, and Yang-Mills Amplitudes V.P. Nair 97

3.1 Fuzzy Spaces 97

3.1.1 Definition and Construction of HN97

3.1.2 Star Products 99

3.1.3 Complex Projective Space CPk 101

3.1.4 Star Products for Fuzzy CPk 103

3.1.5 The Large n-Limit of Matrices 105

3.2 Noncommutative Plane, Fuzzy CP1, CP2, etc 107

3.3 Fields on Fuzzy Spaces, Schr&oddot;dinger Equation 109

3.4 The Landau Problem on R2NC and S2F 111

3.5 Lowest Landau Level and Fuzzy Spaces 113

3.6 Twistors, Supertwistors 115

3.6.1 The Basic Idea of Twistors 115

3.6.2 An Explicit Example 118

3.6.3 Conformal Transformations 118

3.6.4 Supertwistors 119

3.6.5 Lines in Twistor Space 120

3.7 Yang-Mills Amplitudes and Twistors 121

3.7.1 Why Twistors Are Useful 121

3.7.2 The MHV Amplitudes 123

3.7.3 Generalization to Other Helicities 128

3.8 Twistor String Theory 130

3.9 Landau Levels and Yang-Mills Amplitudes 131

3.9.1 The General Formula for Amplitudes 131

3.9.2 A Field Theory on CP1 133

References 135

4 Elements of (Super-) Hamiltonian Formalism A. Nersessian 139

4.1 Introduction 139

4.2 Hamiltonian Formalism 140

4.2.1 Particle in the Dirac Monopole Field 143

4.2.2 K&addot;hler Manifolds 145

4.2.3 Complex Projective Space 146

4.2.4 Hopf Maps 147

4.3 Hamiltonian Reduction 149

4.3.1 Zero Hopf Map: Magnetic Flux Tube 151

4.3.2 1st Hopf Map: Dirac Monopole 152

4.3.3 N+1 &rarrow; PN and T*PN 155

4.3.4 2nd Hopf Map: SU(2) Instanton 156

4.4 Generalized Oscillators 160

4.4.1 Relation of the (Pseudo) Spherical Oscillator and Coulomb Systems 163

4.5 Supersymplectic Structures 167

4.5.1 Odd Super-Hamiltonian Mechanics 171

4.5.2 Hamiltonian Reduction: N+1.M &rarrow; PN.M, ΛN+1 &rarrow; ΛPN 172

4.6 Supersymmetric Mechanics 175

4.6.1 N = 2 Supersymmetric Mechanics with K&addot;hler Phase Space 177

4.6.2 N = 4 Supersymmetric Mechanics 179

4.6.3 Supersymmetric K&addot;hler Oscillator 183

4.7 Conclusion 186

References 186

5 Matrix Models C. Sochichiu 189

5.1 Introduction 189

5.2 Matrix Models of String Theory 190

5.2.1 Branes and Matrices 190

5.2.2 The IKKT Matrix Model Family 191

5.2.3 The BFSS Model Family 192

5.3 Matrix Models from the Noncommutativity 193

5.3.1 Noncommutative String and the IKKT Matrix Model 193

5.3.2 Noncommutative Membrane and the BFSS Matrix Model 199

5.4 Equations of Motion: Classical Solutions 201

5.4.1 Equations of Motion Before Deformation: Nambu-Goto-Polyakov String 201

5.4.2 Equations of Motion After Deformation: IKKT/BFSS Matrix Models 203

5.5 From the Matrix Theory to Noncommutative Yang-Mills 206

5.5.1 Zero Commutator Case: Gauge Group of Diffeomorphisms 206

5.5.2 Nonzero Commutator: Noncommutative Yang-Mills Model 209

5.6 Matrix Models and Dualities of Noncommutative Gauge Models 215

5.6.1 Example 1: The U(1) &rarrow; U(n) Map 217

5.6.2 Example 2: Map Between Different Dimensions 220

5.6.3 Example 3: Change of θ 220

5.7 Discussion and Outlook 223

References 223

Index 227