Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics

Abstract

This book provides an introduction to the cohomology theory of Lie groups and Lie algebras and to some of its applications in physics. The mathematical topics covered include the differential geometry of Lie groups, fibre bundles and connections, characteristic classes, index theorems, extensions of Lie groups and algebras, Chevalley - Eilenberg cohomology of Lie algebras, symplectic cohomology and an introduction to infinite-dimensional Lie groups and algebras. The physical applications include the U(1) (Dirac) monopole, SU(2) instantons and various aspects of anomalies (Wess - Zumino - Witten terms, Abelian and non-Abelian anomaly, path-integral derivation and descent equations). The material presented is essentially self-contained and at a basic graduate text level. The material is also well organized and the book reads very well. The book would be most useful for graduate students and researchers in theoretical and mathematical physics who are interested in applications of Lie group and Lie algebra cohomology in particle physics. Even though most of the proofs of the mathematical theorems are presented, the focus is more on explaining the ideas than striving for mathematical rigour, therefore the book would be less suitable for those students who are interested in the mathematical foundations per se. Since the book seems to aim for physics students, it is a pity that - apart from the topic of anomalies which is covered very thoroughly - the applications are only briefly touched upon. Other applications of current interest, such as non-Abelian monopoles and instantons for gauge groups other than SU(2) as well as their moduli spaces, are not discussed at all even though the necessary mathematical background is presented and thus they seem well within the scope of the book. It must be said, however, that the lack of different applications is compensated for by an excellent set of bibliographical notes and references at the end of each chapter. The book is warmly recommended.