Differential geometry and mathematical physics. II: Fibre bundles, topology and gauge fields (Q313387)

This book is the second part of a two-volume series on differential geometry and mathematical physics, for Part I see [the authors, Differential geometry and mathematical physics. I: Manifolds, Lie groups and Hamiltonian systems. Dordrecht: Springer (2013; Zbl 1259.53003)]. In the first five chapters, general topics of advanced differential geometry -- in particular, fibre bundles -- and analysis on manifolds are studied; including connections, Hodge theory, homotopy theory, the Whitney sum formula, de Rham cohomology, Dirac operators and Sobolev spaces on manifolds, and the Atiyah-Singer index theorem. The remaining four chapters are devoted to the ultimate goal of the series, gauge theory; topics include geometric models for gauge theories and matter fields, the Yang-Mills equation, the Seiberg-Witten model, gauge orbit stratification, and Faddeev-Popov path integral quantization. The latter chapters contain results by the authors on the classification of gauge orbit types and on non-perturbative quantum gauge theory. Additional background on relevant topics in algebraic topology and analysis on manifolds is provided in the appendix. The book is written in the style of a mathematical textbook. The treatment is modern, extensive and rigorous, and classical topics and results of differential geometry (e.g., geodesics, the Gauß-Bonnet theorem) are presented as special cases and corollaries of that general theory. Similarly, classical topics of theoretical physics (e.g., the standard model of elementary particle physics) are placed in a general geometric context and language. The book is addressed to scholars and researchers in differential geometry and mathematical physics, as well as to advanced graduate students who have studied the material covered in the first part of the series. The reader will benefit from remarks and examples in the text, and from the substantial number of exercises at the end of each section.