*(English)*Zbl 1228.81005

This book is the third volume of a complete exposition of the important mathematical methods used in modern quantum field theory. It presents the very basic formalism, important results, and the most recent advances emphasizing the applications to gauge theory. Here, Zeidler concentrates on the classical aspects of gauge theory, i.e., those related to curvature.

The material is divided in four parts. Part I deals with the “Euclidean manifold as a paradigm”. It has twelve Chapters and needs 700 pages to explain relevant algebraic structures, representations of symmetry groups, harmonic analysis, and differential forms. Part II, “Ariadne’s threat in gauge theory”, treats the electromagnetism as a $U\left(1\right)$-gauge theory and the Yang-Mills field as a $U\left(n\right)$-gauge theory. Part III gives an account of “Einstein’s theory of special relativity”, while Part IV explains the formalism and use of cohomology, in particular the de Rham cohomology in connection with the electromagnetic field. Perhaps the book’s greatest strength is Zeidler’s zeal to help students understand fundamental mathematics better. I thus find the book extremely useful since it signifies the role of mathematics for the road to reality (R. Penrose). The main idea is to describe all fundamental forces of nature by the curvature of appropriate fiber bundles.

For a review of the first volume see [(2006; Zbl 1124.81002)]; for a review of the second volume see [(2009; Zbl 1155.81005)].