Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) [Paperback]

Mikio Nakahara (Author)

Book Description

Publication Date: June 4, 2003 | ISBN-10: 0750306068 | ISBN-13: 978-0750306065 | Edition: 2

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.

Product Details

• Paperback: 596 pages
• Publisher: Taylor & Francis; 2 edition (June 4, 2003)
• Language: English
• ISBN-10: 0750306068
• ISBN-13: 978-0750306065
• Product Dimensions: 8.9 x 6.3 x 1.2 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (13 customer reviews)
• Amazon Best Sellers Rank: #588,267 in Books (See Top 100 in Books)

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78 of 82 people found the following review helpful:

5.0 out of 5 stars Best in its genre, September 16, 2002

By 

"jjf-sp" (East Coast, USA) - See all my reviews

This review is from: Geometry, Topology, and Physics

I suppose I should preface this by saying that I read this book *after* reading similar books, so my ability to understand this book is probably better than others, but that said, I think that my comparative evaluation is free from this bias...

There seem to be a few books on the market that are very similar to this one: Nash & Sen, Frankel, etc. This one is at the top of its class, in my opinion, for a couple reasons:

(1) It's written like a math text that covers physics-related material, not a book about mathematics for physicists. I prefer this; you may not. As a consequence, this book is more rigorous than its alternatives, it relies less on physical examples, and it cuts out a lot of lengthy explanation that you may not need. Of course, there are drawbacks to all of these "features" -- you need to decide what you need and what's best for you.

(2) It's most comprehensive, with Frankel coming in second, and Nash & Sen least comprehensive (though they have quite a bit on Fibre bundles and related topics). Nakahara has a chapter on complex manifolds, which is absent from the other two. Nakahara also concludes with a nice intro to string theory, which is absent from the other two as well (though nothing you couldn't find in Polchinski or the like). Actually -- I modify this slightly. Frankel covers less subjects than Nakahara, but with more depth (though also more wordiness -- I quit Frankel about 2/3 through because it wasn't succinct enough and I got tired of it).

Depending on your tastes, I would recommend this book before the other two.

It presupposes that you have an understanding of algebra (groups, rings, fields, etc.) but it has an introduction to the necessary components of topology within. Frankel has presupposes both algebra and topology; Nash & Sen presupposes only algebra.

62 of 65 people found the following review helpful:

5.0 out of 5 stars A wonderful exposition on the mathematics of modern physics, July 22, 2001

By 

hsurreal (Stanford, CA United States) - See all my reviews

This review is from: Geometry, Topology and Physics (Graduate Student Series in Physics) (Paperback)

Nakahara is one of my favorite books. It gives the reader the necessary knowledge in differential geometry and topology to understand theoretical physics from a modern viewpoint. Each chapter in Nakahara would normally take a full semester mathematics course to teach, but the necesseties for a physicist are distilled with just the right amount of rigor so that the reader is neither bored from excessive proof nor skeptical from simple plausibility arguments.

The first few chapters (homotopy, homology) are rather dry, but the text picks up after that. The manifold chapter is really good, particularly the Lie groups section which gives a geometric viewpoint of the objects which get very little attention in a typical particle physics course. Unfortunately, nothing is said on representation theory, but that can be found in Georgi's book. The cohomology chapter is wonderfully quick and to the point. I found myself having to tell myself to slow down because of the excitement I had in reading it. The Riemannian geometry chapter reads wonderfully and serves as a great reference for all those general relativity formulae you always forget. The end of that chapter has an exquisite little bit on spinors in curved spacetime. The complex geometry chapter is also wonderful. I find myself going back to it all the time when reading Polchinski's string text. The chapters on fiber bundles seem a bit on the overly mathy side, but then again, all the pain is in the definitions which becomes well worth it in the end. I haven't read the last few chapters (spending all of my time in Polchinski!) but I definitely will when I have some spare time. The notation in Nakahara is also really self explanatory and standard. It is written with the physicist in mind who doesn't mind a bit of sloppiness or ambiguity in his notation.

With regards to Frankel, Nakahara is much more modular than Frankel. Each chapter of Nakahara is pretty much self contained whereas Frankel kinda needs to be read straight through. I find it very difficult to just look up a random thing in Frankel and learn about it on the spot, whereas this seems to work in Nakahara just fine. Frankel is a bit more respectful of proper mathematics which also makes it a harder text to read for physicists.

Nakahara is a great text. When I visited Caltech I noticed it on the bookshelf of every theorist that I talked to. Anyone who wants to understand how it is that geometry is so important in modern theoretical physics would do himself a favor in buying this book.

23 of 24 people found the following review helpful:

5.0 out of 5 stars Great book., March 25, 2005

By 

Gargantua (France) - See all my reviews

This review is from: Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) (Paperback)

This is a very useful book for understanding modern physics. You absolutely need such a book to really understand general relativity, string theory etc. For instance, Wald's book on general relativity will make much more sense once you go through Nakahara's book. It is very complete, clearly written, comprehensive and easy to read. I would also recommend Morita's "Geometry of differential forms' and Dubrovin,Novikov and Fomeko's 3 volume monograph, if you can find it. All in all, Nakahara's book is one of the best buys, precious book.