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Lectures on fuzzy and fuzzy SUSY physics

Author: A P Balachandran; S Kürkçüoğlu; S Vaidya
Publisher: Singapore ; Hackensack, NJ : World Scientific, 2007.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Starting with the construction of fuzzy spaces, using examples of fuzzy sphere and fuzzy complex projective spaces, this book moves on to discuss the technology of star products on noncommutative R2d  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: A P Balachandran; S Kürkçüoğlu; S Vaidya
ISBN: 9812704663 9789812704665
OCLC Number: 137289528
Description: xiii, 181 pages ; 24 cm
Contents: 2 Fuzzy Spaces 5 --
2.1 Fuzzy C[superscript 2] 5 --
2.2 Fuzzy S[superscript 3] and Fuzzy S[superscript 2] 5 --
2.3 The Fuzzy Sphere S[Characters not reproducible] 5 --
2.4 Observables of S[Characters not reproducible] 8 --
2.5 Diagonalizing L[superscript 2] 9 --
2.6 Scalar Fields on S[Characters not reproducible] 9 --
2.7 The Holstein-Primakoff Construction 10 --
2.8 CP[superscript N] and Fuzzy CP[superscript N] 11 --
2.9 The CP[superscript N] Holstein-Primakoff Construction 14 --
3 Star Products 17 --
3.2 Properties of Coherent States 19 --
3.3 The Coherent State or Voros *-Product on the Moyal Plane 20 --
3.4 The Moyal *-Product on the Groenewold-Moyal Plane 23 --
3.4.1 The Weyl Map and the Weyl Symbol 23 --
3.5 Properties of *-Products 24 --
3.5.1 Cyclic Invariance 24 --
3.5.2 A Special Identity for the Weyl Star 25 --
3.5.3 Equivalence of *c and *w 25 --
3.5.4 Integration and Tracial States 26 --
3.5.5 The [Theta]-Expansion 27 --
3.6 The *-Products for the Fuzzy Sphere 28 --
3.6.1 The Coherent State *-Product *c 28 --
3.6.2 The Weyl *-Product *w 31 --
4 Scalar Fields on the Fuzzy Sphere 35 --
4.1 Loop Expansion 37 --
4.2 The One-Loop Two-Point Function 39 --
4.3 Numerical Simulations 43 --
4.3.1 Scalar Field Theory on Fuzzy S[superscript 2] 45 --
4.3.2 Gauge Theory on Fuzzy S[superscript 2] x S[superscript 2] 46 --
5 Instantons, Monopoles and Projective Modules 47 --
5.1 Free Modules, Projective Modules 47 --
5.2 Projective Modules on A = C[Infinity] (S2) 49 --
5.3 Equivalence of Projective Modules 51 --
5.4 Projective Modules on the Fuzzy Sphere 54 --
5.4.1 Fuzzy Monopoles and Projectors P[Characters not reproducible] 54 --
5.4.2 The Fuzzy Module for the Tangent Bundle and the Fuzzy Complex Structure 56 --
6 Fuzzy Nonlinear Sigma Models 59 --
6.2 CP[superscript 1] Models and Projectors 60 --
6.3 An Action 63 --
6.4 CP[superscript 1]-Models and Partial Isometries 65 --
6.4.1 Relation Between P([superscript [Kappa]]) and P[subscript [Kappa]] 68 --
6.5 Fuzzy CP[superscript 1]-Models 69 --
6.5.1 The Fuzzy Projectors for [Kappa]> 0 69 --
6.5.2 The Fuzzy Projectors for [Kappa] <0 70 --
6.5.3 The Fuzzy Winding Number 71 --
6.5.4 The Generalized Fuzzy Projector: Duality or BPS States 71 --
6.5.5 The Fuzzy Bound 71 --
6.6 CP[superscript N]-Models 73 --
7 Fuzzy Gauge Theories 77 --
7.1 Limits on Gauge Groups 78 --
7.2 Limits on Representations of Gauge Groups 79 --
7.3 Connection and Curvature 80 --
7.4 Instanton Sectors 81 --
7.5 The Partition Function and the [Theta]-parameter 82 --
8 The Dirac Operator and Axial Anomaly 85 --
8.2 A Review of the Ginsparg-Wilson Algebra 85 --
8.3 Fuzzy Models 88 --
8.3.1 Review of the Basic Fuzzy Sphere Algebra 88 --
8.3.2 The Fuzzy Dirac Operator (No Instantons or Gauge Fields) 89 --
8.3.3 The Fuzzy Gauged Dirac Operator (No Instanton Fields) 91 --
8.4 The Basic Instanton Coupling 93 --
8.4.1 Mixing of Spin and Isospin 94 --
8.4.2 The Spectrum of the Dirac operator 94 --
8.5 Gauging the Dirac Operator in Instanton Sectors 95 --
8.6 Further Remarks on the Axial Anomaly 96 --
9 Fuzzy Supersymmetry 99 --
9.1 osp(2, 1) and osp(2, 2) Superalgebras and their Representations 100 --
9.2 Passage to Supergroups 106 --
9.3 On the Superspaces 107 --
9.3.1 The Superspace C[superscript 2, 1] and the Noncommutative C[Characters not reproducible] 107 --
9.3.2 The Supersphere S[superscript (3, 2)] and the Noncommutative S[superscript (3, 2)] 108 --
9.3.3 The Commutative Supersphere S[superscript (2, 2)] 108 --
9.3.4 The Fuzzy Supersphere S[Characters not reproducible] 112 --
9.4 More on Coherent States 114 --
9.5 The Action on the Supersphere S[superscript (2, 2)] 116 --
9.6 The Action on the Fuzzy Supersphere S[Characters not reproducible] 119 --
9.6.1 The Integral and Supertrace 119 --
9.6.2 OSp(2, 1) IRR's with Cut-Off N 121 --
9.6.3 The Highest Weight States and the osp(2, 2)-Invariant Action 121 --
9.6.4 The Spectrum of V 122 --
9.6.5 The Fuzzy SUSY Action 124 --
9.7 The *-Products 125 --
9.7.1 The *-Product on S[Characters not reproducible] 125 --
9.7.2 The *-Product on Fuzzy "Sections of Bundles" 127 --
9.8 More on the Properties of K[subscript ab] 129 --
9.9 The O(3) Nonlinear Sigma Model on S[superscript (2, 2)] 131 --
9.9.1 The Model on S[superscript (2, 2)] 131 --
9.9.2 The Model on S[Characters not reproducible] 132 --
9.9.3 Supersymmetric Extensions of Bott Projectors 133 --
9.9.4 The SUSY Action Revisited 134 --
9.9.5 Fuzzy Projectors and Sigma Models 135 --
10 SUSY Anomalies on the Fuzzy Supersphere 137 --
10.1.1 The Fuzzy Sphere 137 --
10.1.2 SUSY 138 --
10.1.3 Irreducible Representations 139 --
10.1.4 Casimir Operators 140 --
10.1.5 Tensor Products 141 --
10.1.6 The Supertrace and the Grade Adjoint 141 --
10.1.7 The Free Action 141 --
10.2 SUSY Chirality 143 --
10.3 Eigenvalues of V[subscript 0] 144 --
10.4 Fuzzy SUSY Instantons 145 --
10.5 Fuzzy SUSY Zero Modes and their Index Theory 146 --
10.5.1 Spectrum of K[subscript 2] 147 --
10.5.2 Index Theory and Zero Modes 149 --
11 Fuzzy Spaces as Hopf Algebras 151 --
11.3 The Group and the Convolution Algebras 154 --
11.4 A Prelude to Hopf Algebras 155 --
11.5 The *-Homomorphism G* [right arrow] S[Characters not reproducible] 159 --
11.6 Hopf Algebra for the Fuzzy Spaces 161 --
11.7 Interpretation 165 --
11.8 The Presnajder Map 166.
Responsibility: A.P. Balachandran, S. Kürkçüoğlu, S. Vaidya.
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