Symplectic Topology and Floer Homology, Volume 1: Symplectic Geometry and Pseudoholomorphic Curves

This is a two-volume book; this review covers both volumes. See here for volume 2.

The adjective “symplectic” is laden with mathematical meaning and is found in a variety of mathematical contexts, as well as physical ones. I first encountered it in what my undergraduate professor, V. S. Varadarajan, referred to as “Weil’s great Acta paper,” this being André Weil’s seminal 1964 publication in Acta Mathematica, “Sur certains groups d’opérateurs unitaires.” In this paper C. L. Siegel’s analytic theory of quadratic forms is explicated in a magnificent and deep manner, with the symplectic group and unitary representation theory featured center stage. Weil develops the analytic and arithmetic theory of what is now called the Weil representation, actually a projective representation, which is generally called the “oscillator representation” by the physicists. It was first introduced (earlier) by David Shale, a pupil of I. E. Segal, in connection with the behavior of particles satisfying Bose-Einstein (as opposed to Fermi-Dirac) statistics. And, indeed, it is in the context of quantum mechanics that we encounter symplectic structure in connection with quantization.

And so it is that symplectic structure is really part and parcel of mechanics (both classical and quantum) and the first volume of the book under review appropriately starts off with a treatment of symplectic geometry in the setting of Hamiltonian dynamics. This first volume is essentially devoted to what the first words of the book’s title convey, i.e. topology in a symplectic context. We discern that we have here one of those marvels of modern geometry in the large sense, as it were: the interplay between physics, differential geometry, and topology. This is the stuff pioneered and championed by such players as Sir Michael Atiyah, Graeme Segal, the late Raoul Bott, and of course the tragic figure of Andreas Floer. The presence of these geometers in the game is a clue to another major role played by things symplectic: they are found in fixed point and Morse theory in its broad modern sense, seeing that Floer homology is spectacularly used in regard to the Arnol’d conjecture.

Here is how Yong-Geun Oh, the author of this book, starts off the second volume:

In the 1960s, Arnol’d first predicted the existence of Lagrangian intersection theory … as the intersection-theoretic version of … Morse theory and posed [this] conjecture: the geometric intersection number of the zero section of [the cotangent bundle] for a compact manifold … is bounded from below by the one given by the number of critical points provided by the Morse theory on [the manifold].

Oh continues to describe the developing state of affairs regarding this conjecture and notes that “its cohomological version was proven by [Helmut] Hofer using … the classical variational theory of the action functional.” In due course this approach was supplemented by one involving “[a] broken geodesic approximation of the action functional and the method of generating functions,” the players being Laudenbach and Sikorav — “This replaced Hofer’s complicated technical analytic details by simple, more of less standard Morse theory.” He goes on:

Floer introduced … a general infinite-dimensional homology theory … based on the study of the moduli space of an elliptic equation of … Cauchy-Riemann type that occurs as the L²-gradient flow for the action integral associated with the variational problem … [and] Hofer’s theorem [becomes] a special case of Floer’s (at least up to the orientation problem, which was solved by [none other than Oh himself]).

And then Oh adds even more tantalizing material:

Floer’s construction is applicable … also to … various first-order elliptic systems that appear in low-dimensional topology, e.g., the anti-self-dual Yang-Mills equation and the Seiberg-Witten monopole equation, and has been a fundamental ingredient in recent developments in low-dimensional topology as well as in symplectic topology …

So there it is: a solid raison d’être for this scholarly work, and, indeed, a very appealing invitation to this relatively young subject at the intersection of geometry and physics, happily presented by a mathematician to mathematicians. There is so much to be had here, too:

Since the advent of Floer homology in the late 1980s, it has played a fundamental role in the development of symplectic topology … [however] the subject has been quite inaccessible to beginning graduate students and researchers coming from other areas of mathematics … partly because there is no existing literature that systematically explains the problems of symplectic topology, the analytical details and the techniques involved in applying the machinery embedded in the Floer theory as a whole.

Oh goes on to mention Fukaya’s categorification of Floer homology, Kontsevich’s famous “homological mirror symmetry proposal followed by the development of open string theory of D branes in physics [cf. the wonderful online IAS lectures by Nick Sheridan] … [and also] considerable research into applications of symplectic ideas to various problems in (area preserving) dynamical systems in two dimensions …” In any case, with these riches available, the author’s hope “in writing these two volumes [is] … to remedy the current difficulties to some extent” and present an accessible introduction to this beautiful material, albeit at a (necessarily) high level. He notes that “Parts 2 and 3 of these two volumes [respectively, “Rudiments of pseudo-holomorphic curves” and “Lagrangian intersection Floer homology”] could be regarded as the prerequisites for graduate students or post-docs to read [Lagrangian Intersections and Floer Theory: Anomaly and Obstruction]…” The latter book is a two-volume 2009 AMS-published opus by Fukaya, Oh, Ohta, and Ono, focused on the Fukaya category, Novikov theory, tranversality, and a lot more: pretty austere and even avant garde stuff — in fact, Novikov theory, for example, is yet another departure from classical Morse theory in which, for example, the Morse function is replaced by a suitable closed 1-form.

We find ourselves smack-dab in the middle of very serious and advanced modern geometric singularity analysis, to use a ruefully inadequate phrase. The reader should accordingly bear in mind that in cracking the two volumes currently under review, he is in for some very austere stuff: a good introduction to Morse theory (and there is nothing better than Milnor’s book available, of course) is only the bare start of the story.

That said, Oh does take great pains to do exactly as he said above, namely to write something for graduate students and post-docs, and for mathematicians from other areas, of course, and accordingly Part 1 of the first book is explicitly and expansively devoted to “Hamiltonian dynamics and symplectic geometry” and it is only after more than 100 pages or so that Hofer makes his appearance. Parts 2 and 3 have already been mentioned above, and this leaves Part 4: “Hamiltonian fixed-point Floer homology.” Well, yes, here there is really something of a dénouement to be had: we encounter free loop spaces and the according action by S¹, Conley-Zehnder indices (cf. also the book’s Appendix C), quantum cohomology, and spectral invariants galore. This material attests to the depth of this subject, or set of subjects: this is truly state-of-the-art material, and sits at the intersection of some of the hottest mathematics and physics on the current scene, including homological mirror symmetry (as already mentioned) and Gromov-Witten theory, for example.

Finally, there is something very revealing to be gleaned from Oh’s Appendix C, just cited: it is titled “The Maslov index, the Conley-Zehnder index, and the index formula,” and just the first-mentioned object by itself brings in a host of marvels. The Maslov index is present on the scene as a major player in the development of phase integrals in quantum physics (and even Feynman path integrals: cf. relatively recent work by Robbin and Salamon) as well as in number theory, specifically the theory of the Weil representation; in the latter connection see, e.g., the famous book, The Weil Representation, Maslov Index, and Theta Series, by Gérard Lion and Michele Vergne. And this takes us back to the opening remarks of this review and the work of the redoubtable André Weil: an excellent place to stop.

Suffice it to say that Oh’s two volume set is a very valuable contribution to an exceedingly important and lively part of contemporary mathematics, with deep roots in such apparently disparate subjects as number theory, Morse theory, geometry, dynamical systems, and physics. The book evinces serious scholarship and places commensurate demands on the reader, but this is truly beautiful mathematics and it is all well worth it. 

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