Introduction to Strings and Branes

It is hard for even the purest of mathematicians to keep ignorant of what must certainly be the most spectacular interdisciplinarity in the history of our subject in its modern conception, which, for convenience (and aware of controversy in the choice) I’ll situate with Isaac Newton — after all his annus mirabilis of 1666 saw the birth of calculus and the physics that bears his name, interwoven and intertwined. It is an interesting historical exercise to try to trace the eventual bifurcation of what Newton famously called Philosophia Naturalis into mathematics and physics as autonomous entities. Perhaps the separation should be credited to (or blamed on) number theory, seeing that this Queen of the Sciences, as Gauss so poetically described the subject, was already possessed of an autonomous character in the hands of Pierre de Fermat.

It is also interesting in this context to muse on the deep methodological divide that has always distinguished the art of Mathematics (i.e., the business of proving theorems, courtesy of Euclid et al) from science per se, particularly in the context of such post-Newtonian figures as Euler and Gauss, who made major contributions both to mathematics and to physics. As time marched on, however, this practice of wearing of different hats at different times largely disappeared, as pure mathematics and physics took on their more mature shapes, with the critical question of what constitutes rigor taking the role of discriminator. Still, Riemann, Minkowski, and Hilbert, whom we certainly regards as mathematicians first and foremost, occasionally ventured into theoretical physics: Riemann and Minkowski worked on, for instance, theories of space and time (before Einstein), while Hilbert worked on general relativity at about the same time as Einstein. Another profound player along these lines is Von Neumann, of course, with his youthful work in set theory dramatically complemented by his work on functional analysis in the service of quantum mechanics.

Over the last four centuries, otherwise-pure mathematicians have certainly worked in physics, notwithstanding the evocative and exclusive characterization of pure mathematics by G. H. Hardy, probably the most articulate spokesman for the cause: just recall his observation in A Mathematician’s Apology to the effect that never in his life had he done anything applicable (well, except for Hardy-Weinberg, but he can’t really be blamed for that). In fact, Hardy’s credibility is in part connected to the then-defensible fact that for his specialty of the higher arithmetic, i.e. number theory, applications to science or technology certainly looked to be unthinkable.

Well, as we know now, no such luck: in a demonic irony the Queen herself is now irretrievably prostituted to that sexiest of modern applications: the digital revolution, and even the gorgeous theory of elliptic curves now keeps time with cryptography. C’est la guerre et vive la revolution! (I guess).

It’s really all about the proper interaction between theoretical physics, as a sometime branch of mathematics, if I may be so bold and so imperialistic, and other parts of mathematics proper. The book under review is testimony to a contemporary interplay between (very) theoretical physics and, for lack of a more precise phrase, differential geometry. This interaction has already proven itself to be remarkably fecund. In my own work (in number theory) I had occasion in the last decade to study the work of Tom Bridgeland arising from Michael Douglas’ work on D(irichlet)-branes and, in fact, I have had the pleasure of reviewing a book on this subject in this venue. The present text is obviously connected to that, at least in the sense that D-branes are heavily featured: see, e.g., Chapter 15. But we are mathematicians after all, so before anything else, what’s a brane, and what’s a D-brane? Well, in the earlier book I reviewed, Dirichlet Branes and Mirror Symmetry, we learn on p. 8 that a D-brane involves a submanifold of a Riemannian manifold equipped with a vector bundle and a connection, while in the present book we learn on p. 420 (!) that “[b]osonic p-branes [Say it out loud: a particularly unfortunate choice …] are extended objects that sweep out a (p+1)-dimensional space-time surface, often called the world volume, in a background space-time.” If we think of our space-time a (pseudo?) Riemannian manifold, things start to click, and Dirichlet enters the game courtesy of (what else?) certain physically meaningful boundary conditions.

We are actually travelling in the realm where quantum field theory meets string theory: the chapter (14, “Brane dynamics”) of the book under review starts off with the following characterization: “Quantum field theory has traditionally concerned the quantization of point particles and so far this book has largely been about the classical, and then quantum, behavior of strings. In this chapter we will consider objects that sweep out, as they move through space-time, spatial volumes of dimension greater than 1. We refer to such an object as being a p-brane if its world volume contains time and p spatial coordinates. [Thus, a] 0-brane is just a particle and a 1-brane is just a string.” This truly is a brave new world!

As the preceding quote indicates, the book’s first thirteen chapters (and 400-plus pages) are devoted to laying the required foundation for brane-theory properly so called, and we accordingly encounter a trajectory from point particles (Dirac quantization, BRST, &c.), through classical and quantum strings, light cones, spinors, superstrings (again classical and quantum), and more sophisticated string theory, to nothing less than “supergravity theories in four, ten and eleven dimensions.” Daunting stuff, of course, and certainly requiring a lot of space and time to develop: 400 pages is by no means off target.

Obviously this material has to be absorbed for what is ahead. It amounts to a good and worthwhile course in what is still pretty avant garde science (as well as mathematics). It makes for very interesting reading, or studying, for the committed client: if you wish to go at this material the right way, be ready to doodle all over the (happily pretty broad) margins.

Once the branes are in the house, two full chapters are devoted to them, with D-branes the evident pièce de résistance. After this some more mathematically delectable business is done, dealing with connections to the theory of Lie algebras, string theory symmetries, and finally string interactions. Four appendices are added covering, respectively, Dirac and BRST quantization, light cones and spinors, a bit of Riemannian geometry (“The relationship between S² and the Riemann sphere”), and the classical Lie algebras. The latter two testify to the fact that this is really physics doing mathematics, rather than the other way around; the relationship between the 2-sphere and the Riemann sphere is nothing else that the familiar exploitation of the stereographic projection with a bit of differential geometry added.

Let me say that I really, really like this book, and hope to have the time before too long to sit down for quite a spell and do what I recommended above, namely, doodle all over its margins while trying to learn this stuff. My motives are ultimately arithmetical: I am very interested in the interplay between these themes from theoretical physics with the rest of mathematics, particularly the parts I work in. But West’s Introduction to Strings and Branes will score big points with other segments of the audience as well, including of course folks primarily oriented toward physics and properly unable to resist the siren-song of string theory. In fact, I guess the latter will be more comfortable than I with the book’s style, given that, when all is said and done, I side with Hardy. In fact, I run to Landau: give me that old-time Satz-Beweis formulation any time. West’s book is not written this way: neither Sätze nor Beweise. But I guess reading about physics, even when it’s dripping with cool mathematics, should involve the willingness to read, well, physics — and they just don’t talk or write the way we do. Still, this is, as Feynman might have put it, more of the good stuff.

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