11 of 13 people found the following review helpful:
3.0 out of 5 stars
Fair treatment, May 5, 2001
This review is from: Differential Topology and Quantum Field Theory (Paperback)
This book is written for the theoretical physicist in mind. It is somewhat out-of-date, as there have been many developments in differential topology, such as the Seiberg-Witten theory, since this book was published. However, it might still serve the reader with an introduction to these latter developments. Being just a summary there are no proofs given, and this might annoy the mathematician, but it still could be read profitably by a mathematician if they need a quick introduction to the interplay between physics and differential topology. Indeed, some of the results outlined in the book have been dubbed "physical mathematics" by mathematicians because of the lack of rigour involved in some of the constructions. Quantum field theory is not a subject at all to look for mathematical rigour. Several attempts have been made to put it on a sound mathematical foundation, by these attempts all result in a weakening of its physical predictive power.
The author introduces some basic topological notions in the first chapter, such as homotopy and homology/cohomology groups. He does give a good explanation via the smash product, of how to get a base point in a product space when each factor has a base point. Also, his discussion of H-and coH-spaces is very intuitive and serves the physicist-reader well in developing a functorial mind set. Freedman's and Donaldson's work in 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional Euclidean space has recently been shown to have interesting physical consequences, but the author only devotes a few sentences to exotica. The explicit construction of an exotic structure is of great importance to physical applications, but as of yet only existence results are known.
Physicists are used to dealing with elliptic partial differential equations, and the next chapter discusses these in a more abstract guise: the theory of elliptic operators. These are introduced in the context of vector bundles, as preparation for the Atiyah-Singer index theorem. The locality of pseudo-differential operators sets up the need for Sobolev spaces, and the author does a fairly good job of overviewing the main results.
The concept of a sheaf is introduced in the next chapter, but I think physicists would understand sheaves better if they were introduced via analytic continuation, a procedure that physicists are very well acquainted with. K-theory is also discussed in this chapter and the corresponding stable theory. Physicists have to understand the Bott periodicity theorems when doing functional integration in quantum field theory. Characteristic classes are only briefly treated, and, like all the treatments of this subject, the discussion gives no insight as to why these objects work as well as they do.
The author returns to elliptic operators in the next chapter, where their index theory is discussed. The treatment is too formal. and the reader will have to search the literature for more in-depth discussion.
Algebraic geometry, which has taken on immense importance in string theories and M-theory, is introduced in the next chapter. This chapter might be too quick for the physicist needing an understanding for applicaations in these areas. More concrete examples of varieties and explicit calculations of moduli spaces would have been helpful.
Physicists who have done current algebra will appreciate the next chapter on infinite dimensional groups. The loop group, gauge group, Virasoro group, and the Kac-Moody algebra, of use in conformal field theories and gauge field theories, are given fairly good treatment.
Morse theory, so indispensable in both mechanics and quantum field theory, is discussed in the next chapter. This is probably the best written of the chapters in the book, especially the sections on equivariance and supersymmetry.
Instantons, so important in guage theories and the subsequent quantization via functional integration, are treated in Chapter 8. It is a fairly good discussion, with infinite dimensional critical point theory given emphasis.
Applications to string theory is the subject of the next chapter, but the chapter is far too short to be of much use to someone first entering the field.
The treatment of anomalies in the next chapter is quite good though; the section on Fock space and Gauss's law is one of the best I have seen in the literature. The author explains carefully the origin of the Schwinger term.
Conformal field theories follow in Chapter 9, and the Virasoro algebra again makes its appearance. This is an area that employs more of the "hard" analysis in obtaining results rather than "soft" techniques, so physicists should be fairly comfortable with the discussion.
The last chapter introduces a topic that could fairly be classified as a "quantization of mathematics". The author discusses topological quantum field theories, and it is in this area that I believe the most fascinating constructions in all of mathematical physics take place. These theories have spurred a tremendous amount of research, and the author gives a fairly good overview. The book is a little too overpriced considering the content and the fact that it is a paperback. Such expense is worth it for a self-contained book, but this is not one of these, and must be supplemented by a great deal of outside reading.
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