Topology, Geometry, and Gauge Fields

1. Front Matter

   Pages i-xiii

   PDF

2. Geometrical Background

   • Gregory L. Naber

   Pages 1-52

3. Physical Motivation

   • Gregory L. Naber

   Pages 53-160

4. Frame Bundles and Spacetimes

   • Gregory L. Naber

   Pages 161-205

5. Differential Forms and Integration

   • Gregory L. Naber

   Pages 207-295

6. de Rham Cohomology

   • Gregory L. Naber

   Pages 297-350

7. Characteristic Classes

   • Gregory L. Naber

   Pages 351-405

8. Back Matter

   Pages 407-446

   PDF

This volume is intended to carryon the program initiated in Topology, Geometry, and Gauge Fields: Foundations (henceforth, [N4]). It is written in much the same spirit and with precisely the same philosophical motivation: Mathematics and physics have gone their separate ways for nearly a century now and it is time for this to end. Neither can any longer afford to ignore the problems and insights of the other. Why are Dirac magnetic monopoles in one-to-one correspondence with the principal U(l)­ bundles over S2? Why do Higgs fields fall into topological types? What led Donaldson, in 1980, to seek in the Yang-Mills equations of physics for the key that unlocks the mysteries of smooth 4-manifolds and what phys­ ical insights into quantum field theory led Witten, fourteen years later, to propose the vastly simpler, but apparently equivalent Seiberg-Witten equations as an alternative? We do not presume to answer these questions here, but only to promote an atmosphere in which both mathematicians and physicists recognize the need for answers. More succinctly, we shall endeavor to provide an exposition of elementary topology and geometry that keeps one eye on the physics in which our concepts either arose in­ dependently or have been found to lead to a deeper understanding of the phenomena. Chapter 1 provides a synopsis of the geometrical background we assume of our readers (manifolds, Lie groups, bundles, connections, etc. ).

• Area
• Volume
• gauge theory
• mathematical physics
• topology
• combinatorics

From the reviews:

MATHEMATICAL REVIEWS

"The presentation in the remaining five chapters is enriched by detailed discussions about the physical interpretations of connections, their curves and characteristic classes. I particularly enjoyed Chapter 2 where many fundamental physical examples are discussed at great length in a reader friendly fashion. No detail is left to the reader’s imagination or interpretation. I am not aware of another source where these very important examples and ideas are presented at a level accessible to beginners…The topics covered in this book can be found in many other sources, but the present volume discusses with great care those aspects and notions which are particularly important in gauge theory. Moreover, the presentation is backed by many useful and relevant examples and I am convinced that any beginner in gauge theory will find them very useful."

NZMS NEWSLETTER

"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."

SIAM REVIEW

"Naber writes in a most unpretentious style. His prose is not terse like Rudin’s, but not verbose either. He gives full details to all difficult calculations and shows good judgment in deciding what is difficult versus what is not. This is one way in which a writer demonstrates rapport with his/her readers. Never once has Naber omitted anything out of laziness, under the pretense that it is routine. The book is carefully thought out and lecture-tested account of the subject matter listed earlier. It is rigorous, with an emphasis on the details in the examples. Naber favors examples that deal with concrete spaces and revisits them whenever appropriate…In terms of its ability to teach a subject to the novice, this book ranks right up there with many classics…People who collect classics should considerbuying this one, whether or not they plan to study it chapter by chapter. For someone who plans to compute right along with the examples, this book is a must-buy. Naber’s goal is not to teach a sterile course on geometry and topology, but rather to enable us to see the subject in action, through gauge theory. The book is capable of fulfilling this goal because of Naber’s efforts. He has undertaken the arduous task of researching the broad field with its extensive literature, learning the material himself, class testing it in lectures, and agonizing over the best ways to present it. Amazingly, the fruits of his labor can be had for less than $70, thanks to Springer’s consumer-friendly pricing…[the reviewer] hopes that Naber will continue the scholarly program of bringing exciting mathematics and physics to a level of clarity that is within our reach."

 

REVIEWS OF TOPOLOGY, GEOMETRY, AND GAUGE FIELDS:  FOUNDATIONS

"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."
NZMS NEWSLETTER

"...this book should be very interesting for mathematicians and physicists (as well as other scientists) who ae concerned with gauge theories."
ZENTRALBLATT FUER MATHEMATIK

• Department of Mathematics and Statistics, California State University, Chico, USA

  Gregory L. Naber

Policies and ethics