Supersymmetry for Mathematicians: An Introduction (Courant Lecture Notes)

✍ Scribed by V. S. Varadarajan

• Publisher: American Mathematical Society
• Year: 2004
• Tongue: English
• Leaves: 311
• Series: Courant Lecture Notes 11
• Category: Library

✦ Synopsis

Supersymmetry has been studied by theoretical physicists since the early 1970s. Nowadays, because of its novelty and significance--in both mathematics and physics--the issues it raises attract the interest of mathematicians. Written by the well-known mathematician, V. S. Varadarajan, this book presents a cogent and self-contained exposition of the foundations of supersymmetry for the mathematically-minded reader. It begins with a brief introduction to the physical foundations of the theory, in particular, to the classification of relativistic particles and their wave equations, such as those of Dirac and Weyl. It then continues with the development of the theory of supermanifolds, stressing the analogy with the Grothendieck theory of schemes. Here, Varadarajan develops all the super linear algebra needed for the book and establishes the basic theorems: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. A special feature is the in-depth treatment of the theory of spinors in all dimensions and signatures, which is the basis of all supergeometry developments in both physics and mathematics, especially in quantum field theory and supergravity. The material is suitable for graduate students and mathematicians interested in the mathematical theory of supersymmetry. The book is recommended for independent study. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
1.1. Introductory Remarks on Supersymmetry......Page 10
1.2. Classical Mechanics and the Electromagnetic and Gravitational Fields......Page 11
1.3. Principles of Quantum Mechanics......Page 18
1.4. Symmetries and Projective Unitary Representations......Page 26
1.5. Poincare Symmetry and Particle Classification......Page 32
1.6. Vector Bundles and Wave Equations: The Maxwell, Dirac, and Weyl Equations......Page 46
1.7. Bosons and Fermions......Page 57
1.8. Suipersymmetry as the Symmetry of a Z_2-Graded Geometry......Page 60
1.9. References......Page 61
2.1. Geometry of Physical Space......Page 68
2.2. Riemann's Inaugural Talk......Page 72
2.3. Einstein and the Geometry of Spacetime......Page 75
2.4. Mathematical Evolution of the Concept of Space and Its Symmetries......Page 76
2.5. Geometry and Algebra......Page 81
2.6. A Brief Look Ahead......Page 85
2.7. References......Page 88
3.1. The Category of Super Vector Spaces......Page 92
3.2. The Super Poincare Algebra of Gol'fand-Likhtman and Volkov-Akulov......Page 99
3.3. Conformal Spacetime......Page 104
3.4. The Superconformal Algebra of Wess and Zumino......Page 117
3.5. Modules over a Supercommutative Superalgebra......Page 122
3.6. The Berezinian (Superdeterminant)......Page 125
3.7. The Categorical Point of View......Page 128
3.8. References......Page 133
4.1. The Category of Ringed Spaces......Page 136
4.2. Supermanifolds......Page 139
4.3. Morphisms......Page 147
4.4. Differential Calculus......Page 152
4.5. Functor of Points......Page 159
4.6. Integration on Supermanifolds......Page 161
4.7. Submanifolds: Theorem of Frobenius......Page 166
4.8. References......Page 176
5.1. Prologue......Page 178
5.2. Cartan's Theorem on Reflections in Orthogonal Groups......Page 183
5.3. Clifford Algebras and Their Representations......Page 187
5.4. Spin Groups and Spin Representations......Page 201
5.5. Spin Representations as Clifford Modules......Page 212
5.6. References......Page 217
6.1. Introduction......Page 220
6.2. The Central Simple Superalgebras......Page 221
6.3. The Super Brauer Group of a Field......Page 231
6.4. Real Clifford Modules......Page 236
6.5. Invariant Forms......Page 245
6.6. Morphisms from Spin Modules to Vectors and Exterior Tensors......Page 259
6.7. The Minkowski Signature and Extended Supersymmetry......Page 265
6.8. Image of the Real Spin Group in the Complex Spin Module......Page 271
6.9. References......Page 281
7.1. Super Lie Groups and Their Super Lie Algebras......Page 282
7.2. The Poincare-Birkhoff-Wilt Theorem......Page 288
7.3. The Classical Series of Super Lie Algebras and Groups......Page 298
7.4. Superspacetimes......Page 303
7.6. References......Page 308
Back Cover......Page 311