Synopses & Reviews
Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. Modern differential geometry from the author's perspective is used in this work to describe physical theories of a geometric character without using any notion of calculus (smoothness). Instead, an axiomatic treatment of differential geometry is presented via sheaf theory (geometry) and sheaf cohomology (analysis). Using vector sheaves, in place of bundles, based on arbitrary topological spaces, this unique approach in general furthers new perspectives and calculations that generate unexpected potential applications. Modern Differential Geometry in Gauge Theories is a two-volume research monograph that systematically applies a sheaf-theoretic approach to such physical theories as gauge theory. Beginning with Volume 1, the focus is on Maxwell fields. All the basic concepts of this mathematical approach are formulated and used thereafter to describe elementary particles, electromagnetism, and geometric prequantization. Maxwell fields are fully examined and classified in the language of sheaf theory and sheaf cohomology. Continuing in Volume 2, this sheaf-theoretic approach is applied to Yang-Mills fields in general. The text contains a wealth of detailed and rigorous computations and will appeal to mathematicians and physicists, along with advanced undergraduate and graduate students, interested in applications of differential geometry to physical theories such as general relativity, elementary particle physics and quantum gravity.
Review
From the reviews: "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. ... All in all, the book is well written, and it is recommendable to novices and specialists." (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1185, 2010)
Synopsis
Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. In the early 1990s, the author initiated a new kind of differential geometry in which all the machinary of classical differential geometry can be explained without any notion of smoothness, that enables unexpected potential applicability since anomalies can now be incorporated in the calculations. This was acheived via sheaf theory (geometry) and sheaf cohomology (analysis). "Modern Differential Geometry in Gauge Theories" is a two volume research monograph, which systematically applies his sheaf-theoretic approach to such physical theories as gauge theory. Continuing his point of view, started in the first volume of this work, the author extends the application of his sheaf-theoretic approach to Yang-Mills fields in general. The important topics include cohomological classification of Yang-Mills fields, the geometry of Yang-Mills A-connections and moduli space of a vector sheaf, as well as Einstein's equation in vacuum.This text contains a wealth of detailed and rigorous computations, and will appeal to mathematicians and physicists along with advanced undergraduate and graduate students studying applications of differential geometry to physical theories.
Synopsis
This 2-volume research monograph presents a new kind of modern differential geometry approach to such physical theories as gauge theory. Sheaf theory (geometry) and sheaf cohomology (analysis) are used to explain all the machinary of classical differential geometry without any notion of smoothness. Together they present Maxwell fields, Yang-Mills and general relativity. The text contains a treasure trove of minutely detailed and rigorous computations. This original work will appeal to physicists, mathematicians, graduate and advanced undergraduate students studying applications of differential geometry tor physical theories.
Synopsis
Original, well-written work of interest Presents for the first time (physical) field theories written in sheaf-theoretic language Contains a wealth of minutely detailed, rigorous computations, ususally absent from standard physical treatments Author's mastery of the subject and the rigorous treatment of this text make it invaluable
Table of Contents
General Preface.- Preface to Volume II.- Acknowledgments.- Contents of Volume I.- Part II Yang-Mills Theory: General Theory.- 1 Abstract Yang-Mills Theory.- 2 Moduli Spaces of A-Connections of Yang-Mills Fields.- 3 Geometry of Yang-Mills A-Connections.- Part III General Relativity.- 4 General Relativity, as a Gauge Theory. Singularities.- References.- Index of Notation.- Index.