Topology, Geometry and Gauge fields: Foundations

by Gregory L. Naber

Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.

Genres: Mathematics, Physics

457 pages, Paperback

First published April 24, 1997

Reviews

[Brian Powell · Rating 5 out of 5]

There are several books with different combinations of the words "geometry", "topology", and "physics" in the title that seem to be about anything and everything all at once. Take Nakahara's "Geometry, Topology, and Physics" or Frankel's "Geometry of Physics". These are expansive works that seek to bridge the gap between modern physics and cutting edge modern physics by introducing the advanced math needed for the leap. By "cutting edge modern physics" I mostly mean string theory and quantum gravity, though there is some theoretical condensed matter research that employs some sophisticated low-dimensional topology. These books introduce us to differential geometry, algebraic topology, Lie group theory, and more advanced concepts like fiber bundles and characteristic classes. Think of Nakahara as handing a physicist fancy new tools for exploration.

The trouble is that these tools, at the depth they are introduced, only allow us to get started in these subjects. To go further, to *really* understand, we need to consult the mathematics literature more directly. It might seem strange to refer to this situation as "trouble"; after all, Nakahara isn't claiming to give us the whole story, he merely wants to prepare us for the journey. But just giving us tools isn't preparing us, not really. To advance in these subjects, the lines between physicist and mathematician must blur, and so the physicist, in addition to new techniques, must learn to think like a mathematician. Or, at least play the part. It's a difficult and subtle transition, and it requires more than a book of definitions and examples.

Naber's book, despite the title, is not a grab bag of topics like Nakahara or Frankel. Instead Naber's book has a clear focus: he wants to teach us gauge theory. Not the gauge theory we find in particle physics or QFT books, but rather the mathematical theory that underlies them, a theory that fuses geometry and topology into the strange structure called the fiber bundle. It is the mathematical foundations--the bare essence--of the physicist's conception of gauge theory. And it's all the book is about. Along the way, sure, we learn some homotopy and homology and Lie theory, like Nakahara, but it's all with this end in mind.

And it's a great story, masterfully told. The style is formal, but remarkably engaging. It's not a collection of difficult concepts, but an invitation into a world of greater mathematical maturity and perspective. This is actually a very difficult thing to accomplish, to present difficult ideas in a new pedagogy without watering them down to nothingness, as happens often when one gets to this level of sophistication: there is limit beyond which analogies and waving hands just create mush. Because Naber isn't teaching us everything from Lie derivatives to index theorems, he reaches considerable depth and nuance in his development.

Exercises are interspersed inline with the text, almost always directly relevant to the discussion. The section on Vector-valued 1-forms is a tour de force: we define the Cartan form of a Lie group, then discover that part of the SU(2) Cartan form resembles the vector potential of the Dirac monopole. Naber scratches his head and says, "that's tantalizing". How about the Sp(2) Cartan form? Might there something physical lurking there as well? It's then off to the races, as we derive the form, associate it with the quaternionic Hopf fibration, and find that indeed there *is* something there (I won't say what it is). About half this development is left to us as exercises, so that we can play along. It's truly a wild ride. We reach the promised land in the last chapter, where Naber shows us that this powerful, formal theory can be dressed up in the trappings of the physicist's Yang-Mills gauge theory, like he promised all along.

The best part about this text is that it's the first of two. We can stop here if we're satisfied, or we can go further...

[Lulu]

Geometry and topology for physicists