Quantum cohomology is sexy stuff! As indicated in another review, a formula found in 1994 by Maxim Kontsevich in order to display the associativity of the quantum product also provides a recursive means whereby to count the number of plane curves of degree d passing through 3d – 1 points in general position. How marvelous it is to find so basic and classical a problem (in enumerative geometry) settled with complete finality by such hypermodern methods! This is more than enough already to recommend quantum cohomology to the reader. But any subject whose antecedents include physics and algebraic topology (the site where cohomology prefers to reside, although this is open to argument) is bound to evince a host of other aspects and connections beyond enumerative geometry, and this doubtless increases its appeal.
So it is that the book under review, From Quantum Cohomology to Integrable Systems, by Martin A. Guest, focuses on an analytic instead of enumerative geometric aspects of the subject, namely, the connection with differential equations, principally of Korteweg-De Vries (KdV) type. The latter famous equation, universally known by its initials, is characterized as follows on p. 154 of the book: “…the KdV equation [is] one of the most famous integrable systems or ‘soliton equations’ ever studied. To some extent it overshadows the family to which it belongs: since there is no universally accepted definition of integrable system, as a first approximation one could say that an integrable system is a partial differential equation which is analogous to the KdV equation… There are hundreds of treatments of the KdV… [but] our exposition will emphasize D-modules and analogies with quantum cohomology…”
Thus we get a sense of the larger context in which Guest’s treatment can be fitted: the study of PDE using D-modules, which is to say microlocal analysis in the style of Sato and Kashiwara (who recently published an English version of his famous book on D-modules with the AMS). Given that the standard approach to D-modules and microlocal analysis is steeped in sheaf cohomology (i.e. derived and triangulated categories), as set forth in the beautiful book, Sheaves on Manifolds, by Kashiwara and Schapira, it is particularly pleasing to encounter the present connection to quantum cohomology. It should also be noted that, appropriately, Guest wrote From Quantum Cohomolgy to Integrable Systems in Japan, the home of D-modules à la Sato et al.
On to the content of the book, then. Guest offers it as introduction to a young and growing subject, rooted in the work of Witten and Kontsevich “which showed that the KdV appears in higher genus quantum cohomology theory” (something on which “the experts” are still not in agreement as to “what is behind this”); Guest restricts himself to the already very rich genus zero case, obviously for good reason: there is an awful lot going on here.
The book begins with an overview of cohomology as such, goes on to quantum cohomology, quantum differential equations, linear differential equations, quantum D-modules, and then abstract quantum cohomology. After this development Guest proceeds to integrable systems, focusing on KdV equations, as indicated, and finishes with a return to abstract quantum cohomology, now with an eye toward mirror symmetry. Says the author on p. 245: “…mirror symmetry predicted correctly the number of rational curves in one Calabi-Yau manifold… in terms of the coefficients of power series solutions of a differential equation associated with another Calabi-Yau manifold… Only a few of these numbers were amenable to calculation by algebraic geometric methods, yet the mirror conjecture produced all of them at once.” And then: “Mirror symmetry predicts an independent geometrical meaning for [the aforementioned] differential equations…” Indeed, sexy stuff.
Well-written and reasonably paced, From Quantum Cohomology to Integrable Systems is a good introduction to a rich and fascinating subject that is still in its early stages of evolution. The book is indispensable for mathematicians interested in pursuing these ideas and themes.
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.