Geometry, topology, and physics - Nakahara, Mikio (Graduate student series in physics, 1990)
The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories with which the reader is assumed to be familiar. Chapter 2 introduces the mathematical concepts to be used throughout, namely maps, vector spaces and topology. Chapters 3-8 introduce the more elaborate concepts in geometry and topology - Homology and Homotopy groups, Manifolds, De Rham Cohomology groups, Riemannian Geometry and Complex Manifolds - discussing in the appropriate context the application of these concepts to liquid crystals, superfluid Helium, General Relativity and Bosonic String Theory. Chapters 9-12 unify geometry and topology, with the introduction of fibre bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters of the book are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in Gauge Field Theories and the analysis of Polakov's Bosonic String Theory from the geometrical point of view. Geometry, Topology and Physics is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.