by R A Bertlmann, Oxford University Press, ISBN 01 850762 3, pbk £29.95.
Field theory “anomalies” constitute a long-standing source of physics and mathematics. They have remained fascinating for physicists and mathematicians, as ongoing developments in string and brane theory show.
This book gives a comprehensive description of the many facets of this subject that were known before the mid-1980s. It is essentially self-contained and thus deserves to be called a textbook. Both mathematicians and physicists can learn from this volume.
With a modest knowledge of quantum mechanics, a mathematician can read about the history of the subject: the puzzle of the decay of the neutral pion into two gamma-ray photons; the inconsistencies of the perturbative treatment of gauge theories related to the occurrence of anomalies; the original Feynman graph calculations; and the theoretical constructions that introduced relationships with topology, up to the elementary versions of the index theorem for families.
The physicist will find all of the necessary equipment in elementary topology and differential geometry combined in constructions that are familiar to professional mathematicians. S/he will find thorough descriptions of the algebraic aspects that emerged from perturbation theory, both in the case of gauge theories and in the case of gravity, and an introduction to the way in which they tie up with index theory for elliptic operators and families thereof.
The book reads fluently and is written so clearly that one not only gets an overview of the subject, but also can learn it at an elementary level.
The bibliography is a rather faithful reflection of the physics literature and includes a few basic mathematical references, which give the reader the opportunity to learn more in whichever direction s/he chooses.
As mentioned, the subject is still developing in the direction of new mathematics and, possibly, new physics in the context of strings and branes. One may therefore regret that the book stops around the developments that took place in the mid-1980s.
The book is already more than 500 pages. Since it is essentially self-contained and every topic that is dealt with is described in sufficient detail to allow a non-specialist to get acquainted with it, at least at an elementary level, the mathematical techniques do not go beyond elements of differential geometry, as well as of homology, cohomology and homotopy theory. Generalized cohomology theories, including K-theory, only appear in a phenomenological disguise, in connection with the description of the index theorem for families, in the particular case relevant to gauge theories, but not as mathematical prerequisites.
As a consequence of the principle of maximal perversity, one may expect that physics will exhibit subtle effects describable in terms of the above-mentioned constructions. In such an event, there remains the hope for a corresponding textbook as understandable as this one, possibly written by the same author.