Topology and Condensed Matter Physics

It’s a brave new world. Or maybe it’s a brave new world again. The mathematics of the 20th century saw a bifurcation of sorts, with the poles being pure and applied mathematics. The distinct personae of “pure mathematician” and “applied mathematician” were better defined than before. While the 17th-, 18th-, and 19th century titans could count among their ranks any number of scholars who qualified under both headings —in spades: just consider Newton, Euler, Gauss, Riemann and Poincaré — the pendulum swung the other way in the 20th century when the number of mathematicians doing both pure and applied work diminished throughout much of the mathematical world. This state of affairs, at least as far as pure mathematics is concerned, is perhaps best conveyed by G. H. Hardy’s famous utterance found on p. 49 of his A Mathematician’s Apology (1940):

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow.

This is a bit — but only a little bit — disingenuous, since we do have, after all, the Hardy-Weinberg law in population genetics. But Hardy is certainly a representative par excellence of the pure mathematician. He even published a (fantastic!) introductory text titled, A Course of Pure Mathematics.

Of course, the 20th century sports many other such examples: just throw a dart at the list of members of Bourbaki, for example. In contrast, as Constance Reid makes clear in both Hilbert and Courant in Göttingen and New York, the prevailing style at the Georg August Universität (a.k.a. Göttingen) was ostensibly different in that the leading figures, from Gauss through Hilbert, were consistently keen on representing a full spectrum of mathematics, pure and applied, with physics given a lot of air-play. Carl Ludwig Siegel, for example, in (Bourbaki founder) André Weil’s opinion the world’s premiere mathematician of his age, specialized in both (most branches of) number theory and celestial mechanics, this in sharp contrast to Weil (who ranked himself as second). It is arguable that the most persuasive illustration of this inclusive spirit characterizing Göttingen is the seminal publication, Methoden der Mathematischen Physik, popularly known simply as Courant-Hilbert, after the book’s authors, mainstays of Göttingen in its heyday.

That said, it is not off beam to characterize Göttingen’s broadminded style as anomalous, certainly in the West. The Soviets were institutionally more inclined to we(l)d pure and applied mathematics — just consider that major educational organ of the USSR, the famous Mech-Math at Lemonosov University in Moscow, spawning everyone from Arnol’d, Kolmogorov, Gel’fand and Shafarevich to and Andrei Sakharov.

In any case, it is fair to say that many of us have certainly grown up with a sense of separation of mathematics and physics. But as Gershwin would have it, “It ain’t necessarily so.” In the 21st century (but really already in the last decade or two of the 20th), we find that quantum physics has begun to pay back mathematics, with interest. Low-dimensional topology is rife with constructions, assertions, conjectures, theorems and proofs that owe a great deal indeed to what the physicists have been up to, principally in the area of quantum field theory.

A bit of history may be indicated. Let’s start very briefly with quantum mechanics itself, which, due primarily to von Neumann, we can certainly class as a glorious exercise in functional analysis. I never tire of recommending my favorite book in this regard, namely, Prugovecki’s Quantum Mechanics in Hilbert Space, presenting the theory of unbounded densely-defined operators on a Hilbert space (of states). So, very early on, QM began to pay back its debt to mathematics by enriching functional analysis in the indicated manner. As time went on, QM spawned QED, quantum electrodynamics, with the subject’s early pioneer being Dirac, who was deeply unhappy with what would transpire later (specifically the physicists' trickery of renormalization). Around the late 1940s the fat was in the fire: Julian Schwinger, Snin’ichiro Tomonaga, and Richard Feynman finally emerged to propose two versions of QED that at first looked very different. Feynman was doing things his own way, extending methods developed in his PhD thesis at Princeton, where he reformulated QM in terms of what are now know as his path integrals. The latter are notorious (to a mathematician at least) because the spaces over which they integrate generally don’t permit a well-defined measure. However, Feynman’s approach to QED was remarkable — he magically got all the right numbers: his calculations matched experiments to a tee — but what the hell was he doing? The answer to this deep and difficult question was presently given by a convert from Hardy’s circle of pure mathematics, a scholar who had already done serious work in, among other things, transcendental number theory, and who had joined Bethe at Cornell to do physics: Freeman J. Dyson.

The Feynman version of QED is what is generally proposed as the paradigm for the currently very sexy and very popular subject of quantum field theory, QFT. Much of it is really QED, of course, but there’s a lot more, since QFT is in many ways the chosen way to do so much of modern physics, specifically particle physics. This (finally) brings us to the book under review, given that condensed matter physics is one of the main playgrounds for QFT.

It is probably fair to say that for most otherwise pure mathematicians (strangers to condensed matter physics) the great pull QFT exercises is due to its special case, TQFT, topological quantum field theory: it is here that we meet low-dimensional topology and, in particular, knot theory. TQFT is manifested by a functor from closed manifolds together with their bounding manifolds of one dimension less to linear spaces and special vectors therein, and in something like a homology with additive (\(\oplus\)) behavior replaced with multiplicative (\(\otimes\)) behavior. The axioms are due to Atiyah and Segal; see, for example, Atiyah’s elegant and readable book, The Geometry and Physics of Knots. (There is also a compact but very rich article.) The focus of these expositions by Atiyah is the work of Edward Witten, specifically his famous paper, “Quantum Field Theory and the Jones Polynomial” (Commun. Math. Phys. 121(1989), 351–399), in which he derives the Jones polynomial, the quintessential knot invariant, from Chern-Simons theory attached to the corresponding knot in \(S^3\). This is of course truly striking — as Witten himself puts it in the abstract to the paper just cited:

It is shown that \(2 + 1\) dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized from \(S^3\) to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in \(1 + 1\) dimensions.

All this having been said (with apologies to the reader for excessive zeal and long-windedness), we come to the book under review, edited by two physicists (Bhattacharjee, Bandyopadhyay) and a mathematician (Mj), the latter happily being a specialist in hyperbolic geometry and geometric topology. They offer as the book’s raison d’être that “topology needs to be an essential toolkit for the physicists in general and condensed matter physicists in particular,” and therefore the subsequent collection of eighteen lectures (from a SERC school held in Kolkata, India, in the winter of 2015) is presented as a contribution to the cause. In point of fact the editors are more emphatic than this: “At a time when mathematics is getting sidelined in physics curricula [the indicated SERC school] was an attempt to halt, if not change, the trend.”

Accordingly, the book is primarily aimed at physicist, particularly rookies, and therefore the corresponding prerequisites are modest — for physicists — we mathematicians are typically way behind in this regard. Specifically, the reader is supposed to know QM, “including path integrals” — meaning the Feynman formalism —, statistical mechanics, and solid state physics; however “field theory is not a prerequisite.” Thus, as far as the physics side of the interplay between low-dimensional topology and physics is concerned, the book is only a prelude to deeper things. Knot theory is certainly present between the book’s covers, but the Jones polynomial only gets three mentions in the index. Fair enough, after all, by the editors’ own admission, we are dealing with topology as a toolkit for physicists, not geometric topology per se. Well, what’s in it for us mathematicians, then?

Happily, the answer is “quite a lot.” The book under review presents a broad panorama of topological themes we (should) know and (learn to) love, but they are not presented in the depth one would find in more specialized monographs. For example, differential forms and de Rham cohomology occur in Section 5 of Part I, “Topological Tools,” but the whole chapter is only some thirty pages in length, and the two mainstays just mentioned just span some thirteen pages among the thirty: this is not comparable at all to what we find in Bott-Tu’s classic Differential Forms in Algebraic Topology. It’s not supposed to be, of course, and what we find in the present book is excellent in its own way, as well as spiced in what is for us still a pretty unusual way, i.e. it’s oriented toward physics, and this is virtuous in its own right in this brave new world. After we’ve looked at this sketch, we head to Bott-Tu, if we have a true mathematical conscience.

The preceding appraisal can be analytically continued to pretty much all of the book that deals with mathematics (looking toward physics). The various lecturers hit such cool things as fundamental groups of spheres and group actions when looking at homotopy, Bott periodicity when looking at vector bundles, and (no surprise) \(SU(2)\) and \(SU(3)\) in their short course on group theory.

After all this, however, we get to the book’s Part II, “Physics problems,” and we once again ask the question, what’s in it for us?, but now we get a different answer. Perhaps this can be properly characterized as the book’s main attraction: the reader gets to see a number of themes from geometry and topology in the context of physics applications. While not every one’s cup of tea, it is strong tea, without a doubt. For example, the Gauss-Bonnet theorem is presented in a section on quantum mechanics, by no means its traditional context.

A glance at the book’s table of contents will convey at least in broad terms what kinds of physics are presented to the reader, and, if he is a mathematician, he should discern accordingly. But even at the introductory level intended by the authors and editors of the book, there is clearly a huge amount of Feynman’s “good stuff” here. And it does not hurt to repeat that this work is supposed to be just a start for the reader, presumably a fledgling physicist, and that it therefore provides only an entry into the huge and burgeoning subject of the interplay between physics and topology that is so popular today, for all the right reasons. It is indeed good stuff. Onward.

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