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Mikio Nakahara

Geometry, topology and physics

For those who have studied physics to undergraduate level. the abstract geometrical mathematics of research level theoretical physics might seem like a different language. Geometry, topology and physics by Mikio Nakahara helps to bridge that gap. The book starts with a couple of chapters going over undergraduate level physics and mathematics. This is followed by chapters on homology and homotopy groups. Much of the book looks at topics relating to differentiable manifolds, including Riemannian geometry, complex manifolds and fibre bundles, but with an emphasis on their use in quantum theory rather than general relativity.

The final two chapters look at anomalies in gauge field theories and at bosonic string theory.

The book has plenty of diagrams and examples of how the subject matter relates to physical systems. This book doesn't have too steep a learning curve - I felt that someone who had a sound understanding of undergraduate level physics would find it fairly straightforward to work through the first half of the book, but that after that it would become more challenging. Those looking for a gentler approach might prefer The geometry of physics by Theodore Frankel and move on to Nakahara's book when they want to get on to research level topics.

Product Description
Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

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. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber 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 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, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.