Supersymmetry for mathematicians: an introduction. (English) Zbl 1142.58009

Varadarajan, V. S.

Courant Lecture Notes in Mathematics 11. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 0-8218-3574-2/pbk). vi, 300 p. (2004).

Publisher’s description: 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.
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.

MSC:

58A50	Supermanifolds and graded manifolds
16W55	“Super” (or “skew”) structure
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E99	Lie groups
58C50	Analysis on supermanifolds or graded manifolds
81-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81R20	Covariant wave equations in quantum theory, relativistic quantum mechanics