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This introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles. While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. The goal of this book is to present important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This volume started out as notes in an attempt to help my students in a course on classical differential geometry. Not feeling constrained to use one particular text, I went off on my merry way introducing some of the basic structures of classical differential geometry (standard reference: [30]) that are used in physics. When the students requested specific references to texts, I would say that any one from a standard list of quite excellent texts would be fine. But my approach was not to be found in any one of them. Rather, the students had to search here and there in the literature and then try to piece it all together. So my notes were just that: a piecing together of things well known. However, in the spirit of the famous saying of Feynman (see Ref. [14]) noted above, some considerable part of the development of the subject is left for the reader to do in the exercises. Of course, in that same spirit, the reader should create explicitly all of the material presented here.
Stephen Bruce Sontz

Chapter 2. Basics of Manifolds

Abstract
We start off with the idea of a chart. This is sometimes called a system of coordinates, but we feel, as does Lang (see [32]), that this both obscures the basic idea and impairs the recognition of an immediate generalization to Banach manifolds by introducing scads of unnecessary notation. The idea that Banach spaces provide an appropriate scenario for doing differential calculus goes back at least to the treatise [8] of Dieudonné. An extremely well-written text on smooth manifolds is Lee’s book [33].
Stephen Bruce Sontz

Chapter 3. Vector Bundles

Abstract
Right now we would like to understand how the transition functions T ψ β α give us the tangent bundle. And this in turn will motivate the introduction of vector bundles. It is no exaggeration to say that a clear understanding of just the one example of the tangent bundle suffices to give one a clear understanding of vector bundles in general.
Stephen Bruce Sontz

Chapter 4. Vectors, Covectors, and All That

Abstract
In this chapter we will study the material presented so far, but now in the context of a manifold that is an open subset U in the Euclidean space \(\mathbb{R}^{n}\).
Stephen Bruce Sontz

Chapter 5. Exterior Algebra and Differential Forms

Abstract
We give a brief treatment of these topics. The crucial concept here is that of a differential k-form on a manifold. But before discussing that, we have to present a bit of exterior algebra.
Stephen Bruce Sontz

Chapter 6. Lie Derivatives

Abstract
Lie derivatives are discussed in this brief chapter, which I have included in part because it is so traditional. There are two more concrete reasons for this diversion. The first is that Lie derivatives offer some sort of introduction to the idea behind the Frobenius theorem. The second is that they give us an inadequate way of transporting vectors along curves. Why inadequate? This is a technicality, difficult to describe for now. But what we really need to transport vectors in a “parallel” manner is a connection. This is what a connection will do for us. I also discuss why integral curves are not always so important in physics. But be warned that my point of view here is rather heretical.
Stephen Bruce Sontz

Chapter 7. Lie Groups

Abstract
We will need a bare minimum of results on Lie groups. More than anything, we will be establishing notation and giving examples.
Stephen Bruce Sontz

Chapter 8. The Frobenius Theorem

Abstract
It could well be argued that the previous chapters were like an introductory course in a foreign language. Much attention was devoted to learning the new vocabulary and relating it to the vocabulary in a previously known language. Then there were a lot of exercises to learn how the new vocabulary is used. So that’s the grammar. But, to continue the analogy, there was no poetry. Or to put it into colloquial mathematical terminology, there were no real theorems. That is about to change. The Frobenius theorem is a real theorem.
Stephen Bruce Sontz

Chapter 9. Principal Bundles

Abstract
We now introduce the structures that will be used in our applications to physics. We begin with most of the ingredients for the construction of a vector bundle, but now we will use them to construct something else.
Stephen Bruce Sontz

Chapter 10. Connections on Principal Bundles

Abstract
The topic of this chapter has become standard in modern treatments of differential geometry. The very words of the title have even been incorporated into part of a common cliché: Gauge theory is a connection on a principal bundle. We will come back to this relation between physics and geometry in Chapter 14 But just on the geometry side there has been an impressive amount of results, only a fraction of which we will be able to deal with here. Sometimes we speak of the need to translate geometric terminology into physics terminology. And vice versa. Curiously, there is also a need to translate geometrical terminology developed in one context into geometrical terminology from another context. And that is especially true for this topic. In this regard, the books [4] by Choquet-Bruhat and co-authors and [46] by Spivak are quite helpful references. Also quite readable is Darling’s text [6]. In an effort to keep this chapter as efficient as practically possible, we have not presented all the equivalent or closely related ways of approaching this central topic.
Stephen Bruce Sontz

Chapter 11. Curvature of a Connection

Abstract
In this chapter, the discussion will highlight the reasons behind our saying that a connection is a geometric structure. We begin with the crucial concept of the curvature of a connection. First, we introduce some notation.
Stephen Bruce Sontz

Chapter 12. Classical Electromagnetism

Abstract
An example of a classical field is the electromagnetic field, which is some sort of combination of the electric and magnetic fields. This also turns out to be a motivational example in the physics of a gauge theory.
Stephen Bruce Sontz

Chapter 13. Yang–Mills Theory

Abstract
In this chapter, we try to present the theory as Yang and Mills saw it in [54] some 60 years ago. Rather than develop gauge theory in all its generality and then remark that the Yang–Mills theory is just a special case, we want to show the ideas and equations that these physicists worked with, and see later how the generalization came about. So we will use much of the notation of [54], including a lot of indices, as is the tradition in the physics literature to this day. This will require the reader to understand the operations of raising and lowering indices, sometimes not too lovingly known as “index mechanics.” Even though [54] is only five pages long, we only discuss in detail some of its topics. Nonetheless, our approach is somewhat anachronistic since it is our current understanding of gauge theory that illuminates the present reading of [54]. Also, we use a bit of the language of differential forms to prepare the reader for the next chapter. However, this language is inessential for the strict purpose of this chapter.
Stephen Bruce Sontz

Chapter 14. Gauge Theory

Abstract
If this were a novel, then this chapter would be the climax of the story, because here we present the now-legendary result identifying the gauge fields of physics theory with the principal bundles that have a connection of mathematical theory. But in science the story continues and continues. And to this day we do not know how it will end.
Stephen Bruce Sontz

Chapter 15. The Dirac Monopole

Abstract
Dirac introduced the theory of the magnetic monopole which bears his name in 1931 in [9], quite a few years before the advent of either the Yang–Mills theory or the theory of fiber bundles. However, by 1975, Wu and Yang had learned enough about fiber bundles to reformulate Dirac’s theory in terms of a connection on a principal bundle. See [52].
Stephen Bruce Sontz

Chapter 16. Instantons

Abstract
In this chapter we will find certain solutions of the Yang–Mills equations that were given in the now-classic paper [3] by the four authors Belavin, Polyakov, Schwartz, and Tyupkin. Those solutions were called pseudoparticles in that paper, but this is now generally considered to be antiquated terminology. Rather, they are currently called BPST instantons.
Stephen Bruce Sontz

Chapter 17. What Next?

Abstract
This book leaves off well before arriving at the frontier where modern research is occurring. For those who wish to go that far, there are many possible directions, both in mathematics and in physics. But any way one eventually goes, I highly recommend reading and studying a lot.
Stephen Bruce Sontz

Backmatter

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