Modern differential geometry in gauge theories. Vol. II: Yang-Mills fields
This book is the sequel to [Modern differential geometry in gauge theories. Vol. I: Maxwell fields. Boston, MA: Birkhäuser (2006; Zbl 1116.18006)], continuing the study of gauge theories in the framework of abstract differential geometry. It consists of four chapters. The first three chapters are concerned with the general treatment of Yang-Mills theory. Chapter 1 discusses Yang-Mills fields, Yang-Mills equations, Yang-Mills functionals and their variations along with the cohomological classification of Yang-Mills fields, where the author systematically develops all the necessary abstract-differential geometric machinery such as an \(\mathcal{A}\)-metric and its associated abstract Laplace-Beltrami operator yielding the Dirac-Kähler operator and Green's formula. Chapter 2 is concerned with the moduli space of \(\mathcal{A}\)-connections. Chapter 3 deals with the moduli space of solutions of the so-called Yang-Mills equations, which is shown to be so moderate as to admit classical differential-geometric notions such as a tangent vector and a tangent space. The aim of Chapter 4 is to develop the fundamentals of general relativity within the abstract differential-geometric setting, which may be applied to functions with a large amount of singularities. All in all, the book is well written, and it is recommendable to novices and specialists.
scientific article; zbMATH DE number 2167001
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