Supersymmetry For Mathematicians

by Varadarajan, V. S.

• ISBN13: 9780821835746
• ISBN10: 0821835742
• Format: Paperback
• Copyright: 2004-07-01
• Publisher: Amer Mathematical Society

Summary

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 thetheory, 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 thesuper 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 studentsand mathematicians interested in the mathematical theory of supersymmetry. The book is recommended for independent study. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Table of Contents

Preface	vii
Chapter 1. Introduction	1	(58)
1.1. Introductory Remarks on Supersymmetry	1	(1)
1.2. Classical Mechanics and the Electromagnetic and Gravitational Fields	2	(7)
1.3. Principles of Quantum Mechanics	9	(8)
1.4. Symmetries and Projective Unitary Representations	17	(6)
1.5. Poincaré Symmetry and Particle Classification	23	(14)
1.6. Vector Bundles and Wave Equations: The Maxwell, Dirac, and Weyl Equations	37	(11)
1.7. Bosons and Fermions	48	(3)
1.8. Supersymmetry as the Symmetry of a Z2-Graded Geometry	51	(1)
1.9. References	52	(7)
Chapter 2. The Concept of a Supermanifold	59	(24)
2.1. Geometry of Physical Space	59	(4)
2.2. Riemann's Inaugural Talk	63	(3)
2.3. Einstein and the Geometry of Spacetime	66	(1)
2.4. Mathematical Evolution of the Concept of Space and Its Symmetries	67	(5)
2.5. Geometry and Algebra	72	(4)
2.6. A Brief Look Ahead	76	(3)
2.7. References	79	(4)
Chapter 3. Super Linear Algebra	83	(44)
3.1. The Category of Super Vector Spaces	83	(7)
3.2. The Super Poincaré Algebra of Gol'fand-Likhtman and Volkov-Akulov	90	(5)
3.3. Conformal Spacetime	95	(13)
3.4. The Superconformal Algebra of Wess and Zumino	108	(5)
3.5. Modules over a Supercommutative Superalgebra	113	(3)
3.6. The Berezinian (Superdeterminant)	116	(3)
3.7. The Categorical Point of View	119	(5)
3.8. References	124	(3)
Chapter 4. Elementary Theory of Supermanifolds	127	(42)
4.1. The Category of Ringed Spaces	127	(3)
4.2. Supermanifolds	130	(8)
4.3. Morphisms	138	(5)
4.4. Differential Calculus	143	(7)
4.5. Functor of Points	150	(2)
4.6. Integration on Supermanifolds	152	(5)
4.7. Submanifolds: Theorem of Frobenius	157	(10)
4.8. References	167	(2)
Chapter 5. Clifford Algebras, Spin Groups, and Spin Representations	169	(42)
5.1. Prologue	169	(5)
5.2. Caftan's Theorem on Reflections in Orthogonal Groups	174	(4)
5.3. Clifford Algebras and Their Representations	178	(14)
5.4. Spin Groups and Spin Representations	192	(11)
5.5. Spin Representations as Clifford Modules	203	(5)
5.6. References	208	(3)
Chapter 6. Fine Structure of Spin Modules	211	(62)
6.1. Introduction	211	(1)
6.2. The Central Simple Superalgebras	212	(10)
6.3. The Super Brauer Group of a Field	222	(5)
6.4. Real Clifford Modules	227	(9)
6.5. Invariant Forms	236	(14)
6.6. Morphisms from Spin Modules to Vectors and Exterior Tensors	250	(6)
6.7. The Minkowski Signature and Extended Supersymmetry	256	(6)
6.8. Image of the Real Spin Group in the Complex Spin Module	262	(10)
6.9. References	272	(1)
Chapter 7. Superspacetimes and Super Poincare Groups	273
7.1. Super Lie Groups and Their Super Lie Algebras	273	(6)
7.2. The Poincaré-Birkhoff-Witt Theorem	279	(10)
7.3. The Classical Series of Super Lie Algebras and Groups	289	(5)
7.4. Superspacetimes	294	(5)
7.5. Super Poincaré Groups	299	(1)
7.6. References	299