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Enumerative Geometry and String Theory

Sheldon Katz
American Mathematical Society/Institute for Advanced Study
Publication Date: 
Number of Pages: 
Student Mahtematical Library 32
[Reviewed by
William J. Satzer
, on

String theory is a very contentious topic among physicists now, judging by the heat it has generated in scores of related books, articles and blogs over the last few years. Whatever its ultimate fate in physics, string theory has led to a good deal of new mathematics and an upsurge of interest in enumerative geometry. According to the author, the basic question of enumerative geometry is — very broadly speaking — “How many geometric structures of a given type satisfy a given collection of geometric conditions?” For example, what is the number of intersection points of two planar curves in two-dimensional projective space?

Enumerative Geometry and String Theory is part of the Student Mathematical Library series, in this case published jointly by the American Mathematical Society and the Institute for Advanced Study's Park City Mathematics Institute , where the lectures on which this book is based were first given. The books in the series are intended to address non-standard mathematical topics accessible to talented undergraduates with two or more years of college mathematics.

While many of the books in the SML series would be challenging even for strong undergraduates, this one is over the top. For example, Chapters 4 and 5 are called “crash courses”, and they attempt to cover the basics of topology, manifolds and cohomology in thirty-three pages. Chapter 10 — on mechanics (classical and quantum) — should be called a crash course since it races through a treatment of mechanics based on the action principle in about ten pages. Another indication of the difficulty of the subject matter is the author’s comment that his lectures were heavily attended by graduate students specializing in this area who wished to solidify their knowledge of Gromov-Witten theory.

Where is the author going with this? He clearly loves the subject and writes about it with enthusiasm. It is a little hard to find the thread, but it goes something like this: In string theory, Calabi-Yau manifolds are ubiquitous and key to building models of the universe. A particular kind of Calabi-Yau manifold is a quintic threefold, which is a hypersurface of degree 5 in four-dimensional projective space. The number of rational curves on the quintic threefold plays a part in a particularly important calculation in string theory. So, how many rational curves of degree d are there on a quintic threefold? For degree d = 1 it was known in the 19th century that there are 2875 lines on the general quintic threefold. Values for d = 2 and d = 3 had been found by the early 1990s, but it appeared that the standard techniques could not handle the general case. In 1991 a group of physicists announced a solution to this problem using ideas from string theory. There was no proof, but since then the calculation has been formalized and understood as the Mirror Theorem.

The process of connecting physics and enumerative geometry comes to a conclusion in the final chapter where the author applies quantum cohomology to investigate the enumerative geometry of the complex projective plane. Along the way we get a brief introduction to supersymmetry and topological field theory.

Even top notch undergraduates might feel like they’d been it by a speeding bus by the time they get to the end. The topics flash by very quickly and the thread of the author’s argument is difficult to follow. It would have helped considerably if the author had given better indications of his plan along the way, telling us where he’s going and how he plans to get there.

The author notes that the book is not self-contained. This is something of an understatement. Only a background in calculus and some linear algebra is assumed.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

IAS/Park City Mathematics Institute vii

Preface ix

Acknowledgements xii

Chapter 1. Warming up to Enumerative Geometry 1

Exercises 10

Chapter 2. Enumerative Geometry in the Projective Plane 13

Exercises 25

Chapter 3. Stable Maps and Enumerative Geometry 27

Exercises 41

Chapter 4. Crash Course in Topology and Manifolds 43

Exercises 55

Chapter 5. Crash Course in C Manifolds and Cohomology 57

Exercises 76

Chapter 6. Cellular Decompositions and Line Bundles 77

Exercises 92

Chapter 7. Enumerative Geometry of Lines 95


vi Contents

Exercises 108

Chapter 8. Excess Intersection 111

Exercises 123

Chapter 9. Rational Curves on the Quintic Threefold 125

Exercises 133

Chapter 10. Mechanics 135

Exercises 143

Chapter 11. Introduction to Supersymmetry 145

Exercises 156

Chapter 12. Introduction to String Theory 157

Exercises 171

Chapter 13. Topological Quantum Field Theory 173

Exercises 184

Chapter 14. Quantum Cohomology and Enumerative

Geometry 185

Exercises 196

Bibliography 197

Index 201