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Preface vii
Introduction 1
Fuzzy Spaces 5
Fuzzy C[superscript 2] 5
Fuzzy S[superscript 3] and Fuzzy S[superscript 2] 5
The Fuzzy Sphere S[Characters not reproducible] 5
Observables of S[Characters not reproducible] 8
Diagonalizing L[superscript 2] 9
Scalar Fields on S[Characters not reproducible] 9
The Holstein-Primakoff Construction 10
CP[superscript N] and Fuzzy CP[superscript N] 11
The CP[superscript N] Holstein-Primakoff Construction 14
Star Products 17
Introduction 17
Properties of Coherent States 19
The Coherent State or Voros *-Product on the Moyal Plane 20
The Moyal *-Product on the Groenewold-Moyal Plane 23
The Weyl Map and the Weyl Symbol 23
Properties of *-Products 24
Cyclic Invariance 24
A Special Identity for the Weyl Star 25
Equivalence of *c and *w 25
Integration and Tracial States 26
The [Theta]-Expansion 27
The *-Products for the Fuzzy Sphere 28
The Coherent State *-Product *c 28
The Weyl *-Product *w 31
Scalar Fields on the Fuzzy Sphere 35
Loop Expansion 37
The One-Loop Two-Point Function 39
Numerical Simulations 43
Scalar Field Theory on Fuzzy S[superscript 2] 45
Gauge Theory on Fuzzy S[superscript 2] x S[superscript 2] 46
Instantons, Monopoles and Projective Modules 47
Free Modules, Projective Modules 47
Projective Modules on A = C[Infinity] (S2) 49
Equivalence of Projective Modules 51
Projective Modules on the Fuzzy Sphere 54
Fuzzy Monopoles and Projectors P[Characters not reproducible] 54
The Fuzzy Module for the Tangent Bundle and the Fuzzy Complex Structure 56
Fuzzy Nonlinear Sigma Models 59
Introduction 59
CP[superscript 1] Models and Projectors 60
An Action 63
CP[superscript 1]-Models and Partial Isometries 65
Relation Between P([superscript [Kappa]]) and P[subscript [Kappa]] 68
Fuzzy CP[superscript 1]-Models 69
The Fuzzy Projectors for [Kappa] > 0 69
The Fuzzy Projectors for [Kappa] < 0 70
The Fuzzy Winding Number 71
The Generalized Fuzzy Projector: Duality or BPS States 71
The Fuzzy Bound 71
CP[superscript N]-Models 73
Fuzzy Gauge Theories 77
Limits on Gauge Groups 78
Limits on Representations of Gauge Groups 79
Connection and Curvature 80
Instanton Sectors 81
The Partition Function and the [Theta]-parameter 82
The Dirac Operator and Axial Anomaly 85
Introduction 85
A Review of the Ginsparg-Wilson Algebra 85
Fuzzy Models 88
Review of the Basic Fuzzy Sphere Algebra 88
The Fuzzy Dirac Operator (No Instantons or Gauge Fields) 89
The Fuzzy Gauged Dirac Operator (No Instanton Fields) 91
The Basic Instanton Coupling 93
Mixing of Spin and Isospin 94
The Spectrum of the Dirac operator 94
Gauging the Dirac Operator in Instanton Sectors 95
Further Remarks on the Axial Anomaly 96
Fuzzy Supersymmetry 99
osp(2, 1) and osp(2, 2) Superalgebras and their Representations 100
Passage to Supergroups 106
On the Superspaces 107
The Superspace C[superscript 2, 1] and the Noncommutative C[Characters not reproducible] 107
The Supersphere S[superscript (3, 2)] and the Noncommutative S[superscript (3, 2)] 108
The Commutative Supersphere S[superscript (2, 2)] 108
The Fuzzy Supersphere S[Characters not reproducible] 112
More on Coherent States 114
The Action on the Supersphere S[superscript (2, 2)] 116
The Action on the Fuzzy Supersphere S[Characters not reproducible] 119
The Integral and Supertrace 119
OSp(2, 1) IRR's with Cut-Off N 121
The Highest Weight States and the osp(2, 2)-Invariant Action 121
The Spectrum of V 122
The Fuzzy SUSY Action 124
The *-Products 125
The *-Product on S[Characters not reproducible] 125
The *-Product on Fuzzy "Sections of Bundles" 127
More on the Properties of K[subscript ab] 129
The O(3) Nonlinear Sigma Model on S[superscript (2, 2)] 131
The Model on S[superscript (2, 2)] 131
The Model on S[Characters not reproducible] 132
Supersymmetric Extensions of Bott Projectors 133
The SUSY Action Revisited 134
Fuzzy Projectors and Sigma Models 135
SUSY Anomalies on the Fuzzy Supersphere 137
Overview 137
The Fuzzy Sphere 137
SUSY 138
Irreducible Representations 139
Casimir Operators 140
Tensor Products 141
The Supertrace and the Grade Adjoint 141
The Free Action 141
SUSY Chirality 143
Eigenvalues of V[subscript 0] 144
Fuzzy SUSY Instantons 145
Fuzzy SUSY Zero Modes and their Index Theory 146
Spectrum of K[subscript 2] 147
Index Theory and Zero Modes 149
Final Remarks 149
Fuzzy Spaces as Hopf Algebras 151
Overview 151
Basics 153
The Group and the Convolution Algebras 154
A Prelude to Hopf Algebras 155
The *-Homomorphism G* [right arrow] S[Characters not reproducible] 159
Hopf Algebra for the Fuzzy Spaces 161
Interpretation 165
The Presnajder Map 166
Bibliography 169
Index 179
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Add Lectures on Fuzzy and Fuzzy Susy Physics, Noncommutative geometry provides a powerful tool for regularizing quantum field theories in the form of fuzzy physics. Fuzzy physics maintains symmetries, has no fermion-doubling problem and represents topological features efficiently. These lecture notes, Lectures on Fuzzy and Fuzzy Susy Physics to the inventory that you are selling on WonderClubX
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Add Lectures on Fuzzy and Fuzzy Susy Physics, Noncommutative geometry provides a powerful tool for regularizing quantum field theories in the form of fuzzy physics. Fuzzy physics maintains symmetries, has no fermion-doubling problem and represents topological features efficiently. These lecture notes, Lectures on Fuzzy and Fuzzy Susy Physics to your collection on WonderClub |