Geometry, Topology and Physics

by M. Nakahara

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.

Genres: Physics, Mathematics, Textbooks, Geometry, Science, Stem

596 pages, Paperback

First published June 1, 1999

Reviews

[Pratik Rath · Rating 5 out of 5]

I found it the best way to update myself with all the math background I needed to understand modern physics, in particular high energy and condensed matter physics. Useful if one starts covering this foundational stuff in undergrad itself, I personally did so during my Masters. Some of the later chapters are String Theory specific, but atleast upto the chapter of Fibre bundles and connections is very useful for someone working with General Relativity and Gauge Theories.

[Nazir · Rating 3 out of 5]

Went through the first half right up till Riemannian geometry. Depending on what type of learner you are, this may be your cup of tea. To me, there are less painful ways to learn this content (Gauge Fields Knots and Gravity by John Baez for example gave me what I wanted to know about these types of geometry and topology topics). If you’re a physicist who at heart is really a mathematician, this is your book. In my opinion, Manifolds Tensors and Forms by Paul Renteln is like a slightly less painful version of this. This book was pure torture (at least trying to learn from it at first glance). The first 3-4 chapters were okay, the rest was unbearable. I’d give it 2.5/5 stars, but will round up to 3. Just to be nice, and also to objectively appreciate the amount of detail and care Nakahara placed into these topics. Very all encompassing. Just not my speed.

[Israeliano · Rating 3 out of 5]

If you don't know about manifolds, characteristic classes, fibre bundles and index theorems, this book won't help you much. You will drown in a sea of definitions and theorems with little to no "intuition" nor clarification. If you already know any of those topics, but only superficially, this book will help you to go deeper into the topics, although you would need to look somewhere else for an update in many of those topics.

If you are a first-comer to gauge theories as fibre bundles and index theorems, Ooguri's youtube lectures is a better place to update yourself

[Superconformal Hassaan · Rating 5 out of 5]

It is a book that every physics student should read if he/she wants to do work on string theory and/or non perturbative quantum field theory. I will write the rest of the review when I get free.

[Florian Josef · Rating 5 out of 5]

Changed my life

[João · Rating 4 out of 5]

This is one of the best mathematics for physicists books I have ever read. It is targeted at graduate students who want to do research in more theoretical/mathematical aspects of quantum field theory or high energy physics, however, very little mathematical (or physical) details are actually assumed. I myself was quite familiar with the physics but had very little mathematical background (e.g. no topology) and still managed to follow the book with relative ease.

The explanations are incredibly clear throughout the book. Even for the content I was already familiar with I found it quite helpful as it solidified much of my knowledge. For most of the more abstract definitions there is a very clear simple example that makes it a lot easier to follow the slew of definitions (e.g. at the start of "Manifolds" and later of "Fibre Bundles"). And there is a very good balance between interesting/useful proofs that are done explicitly and results that are just quoted without proof (which is a very hard balance to achieve).

The chapter I struggled with most was chapter 10 on "Connections on Fibre Bundles" which I found lacking those kinds of examples I complimented earlier so it was hard to grasp the intuition behind the construction. Additionally, on a few occasions it is a bit encyclopaedic just listing properties and theorems and some of the physical examples at the end of a few chapters lack the context that would make them more rewarding.

All in all, despite a couple of flaws. it was extremely useful in solidifying my mathematical background and I would highly recommend it for graduate students in theoretical/mathematical physics.

[Raphalefrog · Rating 5 out of 5]

A must-have for every mathematical and particle physicist.

[Unpil · Rating 5 out of 5]

A masterpiece that shows a beautiful connection between geometry and physics. This book illuminated to me the elegance behind general relativity and gauge field theories. I am so eager to check out the third edition that is scheduled to come out in mid-2021.