Supersymmetry for Mathematicians: An Introduction (Courant Lecture Notes)

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. This Book Presents The Foundations Of Supersymmetry To The Mathematically Minded Reader In A Cogent And Self-contained Manner. It Begins With A Brief Introduction To The Physical Foundations Of The Theory, Especially The Classification Of Relativistic Particles And Their Wave Equations, Such As The Equations Of Dirac And Weyl. It Then Continues The Development Of The Theory Of Supermanifolds Stressing The Analogy With The Grothendieck Theory Of Schemes. All The Super Linear Algebra Needed For The Book Is Developed Here And The Basic Theorems Are Established: Differential And Integral Calculus In Supermanifolds, Frobenius Theorem, Foundations Of The Theory Of Super Lie Groups, And So On. An Special Feature Of The Book Is The Treatment In Depth Of The Theory Of Spinors In All Dimensions And Signatures, Which Is The Basis Of All Developments Of Supergeometry Both In Physics And Mathematics, Especially In Quantum Field Theory And Supergravity.--jacket. Chapter 2. The Concept Of A Supermanifold 59 -- Chapter 3. Super Linear Algebra 83 -- Chapter 4. Elementary Theory Of Supermanifolds 127 -- Chapter 5. Clifford Algebras, Spin Groups, And Spin Representations 169 -- Chapter 6. Fine Structure Of Spin Modules 211 -- Chapter 7. Superspacetimes And Super Poincare Groups 273. V.s. Varadarajan. Includes Bibliographical References. Supersymmetry has been the object of study by theoretical physicists since the early 1970's. In recent years it has attracted the interest of mathematicians because of its novelty, and because of significance, both in mathematics and physics, of the main issues it raises. This book presents the foundations of supersymmetry to the mathematically minded reader in a cogent and self-contained manner. It begins with a brief introduction to the physical foundations of the theory, especially the classification of relativistic particles and their wave equations, such as the equations of Dirac and Weyl. It then continues the development of the theory of supermanifolds stressing the analogy with the Grothendieck theory of schemes. All the super linear algebra needed for the book is developed here and the basic theorems are established: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. A special feature of the book is the treatment in depth of the theory of spinors in all dimensions and signatures, which is the basis of all developments of supergeometry both in physics and mathematics, especially in quantum field theory and supergravity. "This book presents the foundations of supersymmetry to the mathematically minded reader in a cogent and self-contained manner. It begins with a brief introduction to the physical foundations of the theory, especially the classification of relativistic particles and their wave equations, such as the equations of Dirac and Weyl. It then continues the development of the theory of supermanifolds stressing the analogy with the Grothendieck theory of schemes. All the super linear algebra needed for the book is developed here and the basic theorems are established: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. An special feature of the book is the treatment in depth of the theory of spinors in all dimensions and signatures, which is the basis of all developments of supergeometry both in physics and mathematics, especially in quantum field theory and supergravity."--BOOK JACKET. 1. Introduction......Page 1 2. The concept of a supermanifold......Page 69 3. Super linear algebra......Page 96 4. Elementary theory of supermanifolds......Page 147 5. Spinors......Page 193 6. Super spacetimes and super Poincare groups......Page 305