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Enumerative geometry and string theory. (English) Zbl 1091.14001
Student Mathematical Library 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3687-0/pbk). xiii, 206 p. (2006).
This book grew out of a series of advanced undergraduate lectures given by the author at the Park City Mathematics Institute (PCMI) during its summer program of 2001. As the PCMI (founded in 1991) is to foster interaction between research and education in mathematics by yearly three-week summer programs for researchers and postdoctoral scholars, graduate students, undergraduate students, high school teachers, mathematics education researchers, and undergraduate faculty, and as one of its main goals is to make all of the participants aware of current trends in both mathematics education and research, each summer a different topic is chosen as the focus of discussion. According to this policy, the summer program of 2001 was devoted to recent developments in the interplay of complex geometry and mathematical physics, typified by the spectacular topic “Enumerative Geometry and String Theory”, and the author’s charge was to give students exposure to some of the central ideas, concepts, methods, and results in this current field of research, prudently on an adequate level. In view of this challenging task, the author’s main goals were to provide first an introduction to some basic ideas of classical enumerative geometry, to lead over from there to the rudiments of both modern enumerative algebraic geometry and Gromov-Witten theory, and to explain finally some connections to current topological quantum field theory in physics. However, assuming no specific background knowledge beyond the standard undergraduate courses in mathematics and physics, he has also included some necessary introductory material on abstract algebra, geometry, analysis, and topology, thereby simultaneously broadened the range of areas in undergraduate mathematics for a less seasoned audience of students. As for the incorporation of the relevant physics, the author has skilfully chosen a merely example-driven approach, with emphasizing connections to enumerative geometry throughout. Finally, for the sake of expediency, and in order to streamline the process of this kind of sophisticated undergraduate teaching, sometimes very nonstandard treatments of advanced topics are given, in particular with regard to the involved concepts and methods of modern algebraic geometry. No doubt, this strategy bespeaks the author’s pedagogical passion and mastery, and it is very much to the advantage of non-specialists (or beginners) in the field. The fourteen chapters of the book under review reflect reasonably faithfully both the spirit and the content of the author’s PCMI course on the subject, as he points out in the preface, and the informal classroom style has been retained throughout. Chapter 1 provides a first warming up to enumerative geometry by studying the projective line. Chapter 2 turns then to projective hypersurfaces, with several concrete examples of enumerative problems in the projective plane and an illustration of Bezout’s theorem. Chapter 3 is entitled “Stable maps and enumerative geometry”. By means of the problem of the enumeration of rational curves on a quintic threefold, the author explains the famous Clemens conjecture, the link to mirror symmetry and Gromov-Witten invariants, the notion of stable maps to \(\mathbb{P}^n\) and their naive (compactified) moduli spaces, and the space of plane conics within this framework. In order to delve deeper into this topic, Chapters 4 and 5 are inserted as intermediate crash courses in topology, differentiable manifolds, differential forms, and singular cohomology. Chapter 6 provides more material on cellular homology and cohomology of compact complex manifolds, de Rham cohomology, and line bundles on manifolds.
Chapter 7 returns to enumerative geometry of lines in projective space by discussing Grassmannians, some Schubert calculus, universal bundles on Grassmannians, and vector bundles in general. The example of lines on a quintic threefold is taken up again and analyzed via Chern classes of vector bundles. Chapter 8 extends the explanation of intersection theory (touched upon in Chapter 2) by introducing excess intersection calculations. This is done concretely in the case of plane conics, thereby effectively comparing classical counts and advanced methods in enumerative geometry. Chapter 9 sets up the computation of the number of rational curves on the quintic threefold in the mathematical way, that is by using integration on the moduli spaces \(\overline M(\mathbb{P}^4,d)\) of degree \(d\) and genus 0 stable maps to \(\mathbb{P}^4\). In this chapter, the author also explains how this computation was inspired by intuitive reasonings from topological string theory in physics. This connection serves as the driving motivation for the remaining five chapters, in which the increasing cross-fertilization of enumerative algebraic geometry and theoretical physics is elucidated from a more physical point of view. Chapter 10 gives a brief introduction to classical mechanics and quantum mechanics, touching upon conformal symmetry and Feynman path integrals, whereas Chapter 11 discusses the concept of supersymmetry by means of some examples from 0-dimensional supersymmetric quantum field theories. This is used to illustrate how ideas from physics can be applied to solve enumerative problems in projective geometry. Chapter 12 briefly describes the ideas of bosonic string theory, the fermionic symmetry of the \(A\)-model, and the related BRST cohomology, again illustrated by the example of a quintic threefold as underlying spacetime. The relations to topological quantum field theory are explained in Chapter 13, with a special emphasis on Calabi-Yau threefolds, the B-model, the phenomenon of mirror symmetry, and the axiomatic definition of a \((1+1)\)-dimensional topological quantum field theory. The initial sketch of quantum cohomology given in this context is made more rigorous in the final Chapter 14, where the algebro-geometric approach to Gromov-Witten invariants via integrals on moduli spaces of stable maps is described. Various examples are then given to show how the quantum cohomology of \(\mathbb{P}^2\) can be calculated, and how Gromov-Witten invariants can be applied to the enumerative geometry of of the projective plane. At this point, the author has come full circle, returning to plane enumerative geometry after quite a tour through modern complex geometry and string theory.
Each chapter comes with a number of related exercises, most of which are however quite challenging. It is understood that the ambitious reader will follow the hints for further reading, on the one hand, or seek professional help from teachers or graduate students, on the other. As the author indicates already in the preface, this book will be quite demanding for an undergraduate student, especially with regard to the physics-related chapters at the end. Nevertheless, this text is perfectly suited for giving students a first research experience in a current field of mathematical activity, and it is certainly a lovely invitation to the subject. The author has managed to provide the first mathematically profound down-to-earth introduction to this fascinating area in contemporary mathematics and physics for beginners, and that with admirable didactic mastery. A steadfast student can profit a great deal from working through this text, be it by an impetus for a possible academic career in the future orby getting just a flavour of what is going on in current mathematics and physics. However, this book is by far not self-contained, and it is not meant to be a substitute for a more thorough and more systematic treatment of any of the topics panoramically reviewed here. But it surely is an excellent introduction to the more advanced monographs in the field, among those being [“Mirror symmetry and algebraic geometry” by D. A. Cox and S. Katz (Mathematical Surveys and Monographs, 68, AMS, Providence, RI) (1999; Zbl 0951.14026)] and [“Mirror symmetry” by K. Hori, S. Katz, A. Klemm, R. Pandharipande R.Thomas, C. Vafa, R. Vakil and E. Zaslow (Clay Mathematics Monographs 1, AMS, Providence, RI) (2003; Zbl 1044.14018)].

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
81T45 Topological field theories in quantum mechanics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory