Principal bundles are of huge importance in a number of interrelated subjects in the general areas of differential geometry, algebraic topology, and the representation theory of Lie groups. They can be defined briefly as fiber bundles whose structure group is a Lie group whose associated representation (in its own group of automorphisms) is given by left multiplication; in this connection see p. 68 of *Mathematical Aspects of Quantum Field Theory* by de Faria and de Melo. In the first volume of the set of books under review here, however, the full definition of these objects doesn’t occur until p. 113, after a rather dramatic build-up in differential geometry. The point is that we have here an example (in fact an example *par* *excellence*) of how things are these days when geometry, topology, and physic s supply one another with all sorts of riches.

De Faria and de Melo are keen on sketching a panorama of QFT replete with coverage of classical and quantum mechanics, as well as loads of field theory (in the sense of physics, not algebra, of course), and the particulars of differential geometry occupy just their fourth chapter, “Fiber bundles, connections, and representations.” Sontz, on the other hand, is interested in something completely different. He, too, pays proper homage to both geometry and physics (just look at the titles of the individual volumes under review), but he is out to teach a very detailed course in differential geometry, with principal bundles at its heart, and their roles throughout both mathematical and physical spectra presented with zest. Thus, before he gets to the stated subject of these two books, he presents the reader/student with a true course on manifolds replete with the indicated Lie theory. He goes from manifolds and vector bundles, through exterior algebra, to Lie derivatives, Lie groups and the Frobenius theorem; only after all this, in the middle of the first volume, does Sontz get to principal bundles properly so-called. There is a lot of foreshadowing and motivation, to be sure, but he is careful in dotting all the *i*s and crossing all the *t*s throughout.

After Sontz has brought principal bundles to central stage, he gets to the geometry of modern physics at the speed of light (Couldn’t resist …). The first volume finishes with a very thorough discussion of connections (and their curvature), electromagnetism (including gauge theoretical aspects, both for Maxwell’s equations and for quantum mechanics), Yang-Mills theory, gauge theory proper (as a *Ding in sich*), the Dirac monopole, and finally instantons. Wow!

Well, there’s more, actually, namely the second volume, devoted to “the quantum case,” as opposed to “the classical case,” as per the usual distinction made in modern physics. What then does this quantum case consist in? In Sontz’s hands the answer takes us from Hopf algebras and hyped-up differential calculus, braided exterior algebras, and ~~x~~–structures (in other words, apparently pretty arcane physicists’ stuff) to quantum principal bundles, the *pièce de résistance*.

Leaving aside for a moment the issue of, e.g., what all this tweaking of differential calculus consists in (but see below), what does quantizing principal bundles mean? It turns out that there’s quite a bit involved in it, but Sontz provides on p. 182 that in comparison with classical bundles, “the ‘total space’ is [now] represented by [a] ~~x~~–algebra … which is intuitively a ‘quantum space.’ On the other hand, the structure group of the bundle is represented by [a] ~~x~~–Hopf algebra … which is intuitively a ‘quantum group.’” So we see here a manifestation of a considerably evolved form of the philosophy of quantization which Folland, for example, describes very well indeed on p. 15 of *Harmonic Analysis in Phase Space* as “setting up a correspondence … between classical and quantum observables.” The idea is to tweak classical constructs so as to get a corresponding theory valid for the quantum domain, and here the main idea is to go at differential calculus in a very broad interpretation. Says Sontz on p. 7 of his second volume:

The construction of an adequate differential calculus for a given quantum space (generalizing the de Rham theory in the exterior algebra associated to a smooth manifold) is a nontrivial problem. And the resulting theory was not as one had anticipated and was even, at first, considered to be defective in some intuitive sense. The first step is to construct a generalization of the de Rham differential …

Bingo! We have the rationale for the aforementioned tweaking of differential calculus — see p. 8: it largely consists in changing the notion of a differential form to the effect of requiring its coefficients to come from a special algebra. Down the line, the algebra can have a ~~x~~–structure and we’re off to the races.

*A propos*, here is a very salient quote from p. 3 of the second volume, Sontz’s second introduction:

… some of the original ideas about non-commutative geometry go back to Murray and von Neumann … but it has been the works of Connes and Woronowicz that have given much of the flavor to current research by showing how differential calculi work in the non-commutative world. Also, the introduction of plenty of examples of quantum groups of interest in physics by Faddeev and his collaborators has been another critically important aspect of the modern approach.

And so the cat is out of the bag (with apologies to Schrödinger, I guess): we’re doing some heavy duty non-commutative geometry and find ourselves pretty much at one of the principal frontiers of where geometry, topology, and physics meet.

In closing, Sontz’s two books are very thorough, very sound pedagogically, and, to me, very exciting. He paints a broad but quite detailed picture of some beautiful mathematics and its interface with physics and highlights some of the most important themes in differential geometry as he does so. He is serious about teaching these things well, given his introductions, motivations, explanations, asides, examples, and exercises: the full package. This is an excellent set of books on differential geometry and a lot more. They should be of particular interest to mathematics students and mathematicians needing to know physics done right (from our own parochial point of view).

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.