Modern differential geometry in gauge theories. Vol. II: Yang-Mills fields
type: Item
instance of: scholarly article
title: Modern differential geometry in gauge theories. Vol. II: Yang-Mills fields
zbMATH Open document ID: 1185.18001
DOI: 10.1007/978-0-8176-4634-9
author: Anastasios Mallios
published in: Progress in Mathematical Physics
publication date: 2005-05-10T00:00:00Z
review text: 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.
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18-01
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 53-01
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 81-01
zbMATH DE Number: 2167001
zbMATH Keywords: Yang-Mills field
zbMATH Keywords: Yang-Mills equation
zbMATH Keywords: Yang-Mills functional
zbMATH Keywords: gauge theory
zbMATH Keywords: Green's formula, Dirac-Kähler operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: moduli space
zbMATH Keywords: general relativity
zbMATH Keywords: abstract differential geometry
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-4634-9
OpenAlex ID: W1952594734
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description: scientific article; zbMATH DE number 2167001