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Home , Enumerative geometry, Mathematical analysis, Quantum cohomology, Sheldon Katz

STUDENT MATHEMATICAL LIBRARY http://dx.doi.org/10.1090/stml/032 UlAS/PARK CITY MATHEMATICAL SUBSERIES 32

Enumerative and Sheldon Katz

American Mathematical Society Institute for Advanced Study Editorial Board of the Student Mathematical Library Robin Forman (Chair) Brad Osgood Michael Starbird

Series Editor for the Park City Institute

John Polking

2000 Mathematics Subject Classification. Primary 14N10.

For additional information and updates on this book, visit www.ams.org/bookpages/stml-32

Library of Congress Cataloging-in-Publication Data Katz, Sheldon, 1956- and string theory / Sheldon Katz. p. cm. — (Student mathematical library v. 32) Includes bibliographical references and index. ISBN 0-8218-3687-0 (alk. paper) 1. Geometry, Enumerative. 2. String models. I. Title. II. . QA607.K38 2006 516.3'5—dc22 2006040710

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Dedicated to the memory of Isadore Glaubiger This page intentionally left blank Contents

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 13 Exercises 25

Chapter 3. Stable Maps and Enumerative Geometry 27 Exercises 41

Chapter 4. Crash Course in and 43 Exercises 55

Chapter 5. Crash Course in C°° Manifolds and 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 125 Exercises 133

Chapter 10. Mechanics 135 Exercises 143

Chapter 11. Introduction to 145 Exercises 156

Chapter 12. Introduction to String Theory 157 Exercises 171

Chapter 13. Topological Quantum Theory 173 Exercises 184

Chapter 14. and Enumerative Geometry 185 Exercises 196

Bibliography 197

Index 201 IAS/Park City Mathematics Institute

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the "Regional Geometry Institute" initiative of the National Foundation. In mid-1993 the program found an in• stitutional home at the Institute for Advanced Study (IAS) in Prince• ton, New Jersey. The PCMI continues to hold summer programs in Park City, Utah. The IAS/Park City Mathematics Institute encourages both re• search and education in mathematics and fosters interaction between the two. The three-week summer institute offers programs for re• searchers and postdoctoral scholars, graduate students, undergradu• ate students, high school teachers, research• ers, and undergraduate faculty. One of PCMI's main goals is to make all of the participants aware of the total spectrum of activities that occur in mathematics education and research: we wish to involve pro• fessional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that end the sum• mer institute features general sessions designed to encourage interac• tion among the various groups. In-year activities at sites around the country form an part of the High School Teacher Program.

vn viii IAS/Park City Mathematics Institute

Each summer a different topic is chosen as the focus of the Re• search Program and Graduate Summer School. Activities in the Un• dergraduate Program deal with this topic as well. Lecture notes from the Graduate Summer School are published each year in the IAS/Park City Mathematics Series. Course materials from the Undergraduate Program, such as the current volume, are now being published as part of the IAS/Park City Mathematical Subseries in the Student Mathematical Library. We are happy to make available more of the excellent resources which have been developed as part of the PCMI.

John Polking, Series Editor February 20, 2006 Preface

This book is based on a series of fifteen advanced undergraduate lec• tures I gave at the Park City Mathematics Institute (PCMI) during the summer of 2001. A separate book based on the program of grad• uate lectures is being published in the Park City Mathematics Series. My charge was to give students exposure to some of the important ideas related to modern enumerative geometry as it has evolved in re• cent years. My goals were to provide some background in and inspira• tion from classical enumerative geometry, to explain the rudiments of stable maps and Gromov-Witten theory, and to explain connections to string theory in . The students were quite good, as I had been told to expect. Stu• dents were selected from a competitive application process. Further• more, PCMI runs two series of undergraduate lectures: an "elemen• tary" lecture series and an "advanced" lecture series. The students self-select which lecture series they will attend, so my lectures were populated by the strongest of this already talented of under• graduates. The elementary lectures were given by Ruth Gornet on the subject of . While my experience with the undergraduate students was close to my expectations, there was however a big surprise: my lectures X Preface were heavily attended by graduate students. Graduate students re• peatedly told me that my lectures helped them solidify their knowl• edge of Gromov-Witten theory, as I presented a more leisurely intro• duction than they had seen before, made clearer than they had seen before how Gromov-Witten theory builds upon classical enumerative geometry, gave many examples, and connected to physics more than they had seen before. For this reason, while writing this book pri• marily for advanced undergraduates, I am keeping graduate students in mind as an important secondary audience. Designing the course was a bit of a challenge. I could assume that the students were smart and willing to work hard and that they had "mathematical maturity", but I could not assume exposure to any specific of mathematics beyond a standard undergraduate course in linear . For that reason, the lectures contained introductory material on , geometry, , and topology. Most of the participating students already knew some of these topics, but few knew all of them. Thus the lectures served both as a review and as an introduction to a range of in undergraduate mathematics. The incorporation of physics presented another challenge, as I could not assume anything more than exposure to a first undergrad• uate physics course. Here, I did not even pretend to be pedagog• ical or complete. I cut corners by explaining a range of relevant ideas of physics via the simplest examples, emphasizing connections to enumerative geometry throughout. While the physics lectures were undoubtedly the most difficult part of the course for the students, I hoped that they would get a firm impression of the myriad connections between geometry and physics through this very brief introduction. I have similar hopes for the reader of this book. It has certainly been gratifying to see a number of "my" PCMI undergraduate students currently pursuing graduate studies in this re• search area, near and dear to my own heart. But the PCMI program has a broader purpose—to give students a research experience that will benefit them in their chosen careers. I propose that the model adopted here for undergraduate training—shooting for some reason• ably advanced ideas from graduate level mathematics while filling in Preface XI

a broad range of topics en route—is a good way to help advanced un• dergraduates integrate what they have learned and prepare for their careers. Whether or not they pursue graduate studies in the particu• lar focus areas seems secondary to me, even though I am quite fond of the subject area of this book. Turning the lectures into a book introduced another of chal• lenges. Undergraduate students reading this book will probably not have the benefit of lectures at which they can ask questions as they go along. In addition, the PCMI students were helped by an expert teaching assistant, Artur Elezi. Since I assume that the reader will not have direct access to a faculty member, I have added much back• ground material that was left out of the lectures, while trying to keep the informal feel throughout. I added material on differential geome• try which was not needed in the original lectures since this topic was covered in Ruth Gornet's lectures. Even so, the reader is warned that this book is not self-contained. In particular, working through this book is not a substitute for a more thorough and more pedagogical treatment of any of the background topics reviewed here. I have cer• tainly cut out much important foundational material to streamline the process of getting to the desired results. Even worse, sometimes for the sake of expediency I have given very nonstandard treatments of subjects. Some of these have a somewhat clumsy feel to them. This is especially true of the ad hoc treatment I have given of the beautiful subject of . An undergraduate student interested in algebraic geometry reading this book is urged to consult the refer• ences provided. A graduate student interested in algebraic geometry reading this book is urged to provide the standard definitions of all of the concepts which have been introduced here in an ad hoc fashion. Each of the chapters of the book corresponds reasonably closely to one of the lectures, the exception being that the material from a lec• ture entitled "More on bundles" has been divided up and distributed through other chapters, so that there are only fourteen chapters re• placing fifteen lectures. Since material has been added beyond the content of the original lectures, each chapter contains substantially more than one lecture's worth of material. Xll Preface

As I have indicated above, this book will be quite challenging for an undergraduate student, and, to be honest, will possibly be too hard to understand completely in a few spots, especially the physics- related material near the end. But I encourage all undergraduate students to "go for it" and persist with the difficult parts. If you reach for the stars, you will at least get to the moon. That said, students would likely get more out of the book if they could consult a faculty member or graduate student when needed, as in a directed reading class.

Acknowledgements

During the more than four years that I have worked on this project, I have been helped in many ways by many people. First and fore• most, I would like to thank the students who attended my lectures. They helped with their questions and corrections to the original set of lecture notes, helping me to improve the presentation. More sig• nificantly, their energy and enthusiasm convinced me that it would be worthwhile to expand the lectures into a book. They will unfortu• nately have to remain nameless, as I did not take roll and so would probably miss at least a few dozen if I tried to reconstruct the com• plete list from PCMI records and my memory. You know who you are, and I want to express my thanks to you here. I also owe a debt of gratitude to Artur Elezi, who both served as a Teaching Assistant at PCMI and suggested many improvements to the original lecture notes. I would also like to give special thanks to Michael Mulligan and Michael Sommers. These two University of Illinois students read most of the manuscript at different points, reported numerous typos and mistakes, and offered suggestions for improvement. Michael Mulligan worked through Chapters 1-10 during 2004. Michael Sommers read a near-final draft during the summer of 2005 with particular attention to Chapters 10-14 and provided assistance with the bibliography and the index. The book is much better as a result of their efforts. I also owe a debt of gratitude to Arthur Greenspoon, who volun• teered his professional skills and proofread the entire book. Parts of the book were proofread by Daniel Morton, whom I thank as well. Acknowledgements xm

Undoubtedly a number of typos and other errors are present in the final version, for which I alone bear responsibility. I am maintain• ing a website for the book. It can be reached from the AMS webpage for the book, whose URL is given on the copyright page and on the back cover of the book. The website will contain lists of typographical errors and their corrections. I thank David Cox, Joachim Koch, and Michael Stone for assis• tance with figures, and I also thank Randy McCarthy for providing some references. A number of people involved with PCMI deserve thanks for their assistance: William Barker, Herb Clemens, Catherine Giesbrecht, Roger Howe, and David Morrison. I also thank the AMS staff who have assisted me directly during this project: Barbara Beeton, Arlene O'Sean, and especially Edward Dunne, as well as the rest of the team working behind the scenes to make this book a reality. Finally, I want to thank my family for their and patience during the times when this project made me less available for them than any of us would have liked. November 2005 Sheldon Katz This page intentionally left blank Bibliography

We use the following conventions for labeling references in the text and in the bibliography:

• References with one author are labeled by the author's last name. • References with multiple authors are labeled using the first letter of the authors' last names in alphabetical order. • If there is more than one reference for the same author or group of authors, they are ordered by publication year.

[Ahlfors] L. Ahlfors, , McGraw-Hill, New York, 1978. [Arnold] V. I. Arnold, Mathematical Methods of , Grad. Texts in Math. 60, Springer-Verlag, New York, 1997. [Atiyah] M. Atiyah, The Geometry and Physics of Knots, Lezioni Lincee, Cambridge Univ. Press, Cambridge, 1990. [BT] R. Bott and L. W. Tu, Differential Forms in , Grad. Texts in Math. 82, Spring-Verlag, New York, 1982. [CdOGP] P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory, Nuclear Physics B359 (1991), 21-74 and in Essays on Mirror Mani• folds (S.-T. Yau, ed.), International Press, Hong Kong, 1992, 31-95. [CGGH] J. Carlson, M. Green, P. Griffiths, and J. Harris, variations of Hodge structure, I, Comp. Math. 50 (1983), 109-205.

197 198 Bibliography

[Clemensl] H. Clemens, Homological equivalence modulo algebraic equiva• lence is not finitely generated, Inst. Hautes Etudes Sci. Publ. Math. 58 (1983), 19-38 (1984). [Clemens2] H. Clemens, A Scrapbook of Complex Curve Theory, Grad. Stud. Math. 55, AMS, Providence, RI, 2003. [Conway] J.B. Conway, Functions of One Complex I, 2nd ed., Grad. Texts in Math. 11, Springer-Verlag, New York, 1978. [CK] D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, AMS, Providence, RI, 1999. [ES] G. Ellingsrud and S. A. Str0mme, The number of twisted cubics on the general quintic threefold, Math. Scand. 76 (1995). [Feynmanl] R. P. Feynman, R. Leighton, and M. Sands, The Feynman lectures on Physics, Vols. 1, 2, 3, Addison-Wesley Publishing Co., Inc., Reading, MA, 1963, 1964, 1965. [Feynman2] R. P. Feynman, QED: The Strange Theory of Light and Mat• ter, Press, Princeton, NJ, 1988. [FU] Geometry and (Freed and Uhlenbeck, eds.), IAS/Park City Mathematics Series 1, AMS, Providence, RI, 1995. [Freed] D. Freed, Five lectures on super symmetry, AMS, Providence, RI, 1999. [Fultonl] W. Fulton, Introduction to in Algebraic Ge• ometry, CBMS Regional Conference Series in Mathematics 54, AMS, Providence, RI, 1984. [Fulton2] W. Fulton, Intersection Theory, Springer-Verlag, New York- Berlin-Heidelberg, 1984. [Fulton3] W. Fulton, Algebraic curves, Adv. Book Classics, Addison- Wesley, Redwood City, CA, 1989. [FP] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic Geometry (Santa Cruz, 1995) (J. Kollar, R. Lazarsfeld, and D. Morrison, eds.), Proceedings of Symposia in 62, Part 2, AMS, Providence, RI, 1997, 45-96. [Givental] A. Givental, Equivariant Gromov-Witten Invariants, Internat. Math. Res. Notices (1996), 613-663. [GSW] M. B. Green, J. H. Schwarz, and E. Witten, Superstring theory, Vols. 1 and 2, Cambridge University Press, Cambridge, UK, 1987. [Greenberg] M. Greenberg, Lectures on Algebraic Topology, Benjamin, New York, 1967. [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Bibliography 199

[Harris] J. Harris, Algebraic Geometry: A First Course, Grad. Texts in Math. 133, Springer-Verlag, New York, 1992. [Hartshorne] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York-Heidelberg, 1977. [Hilbert] Mathematical Developments arising from Hilbert problems, F. Browder (ed.), Proceedings of Symposia in Pure Mathematics 28, AMS, Providence, RI, 1976. [HKKPTVVZ] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow. Mirror symmetry, Clay Mathematics Monographs, AMS, Providence, RI, 2003. [Hulek] K. Hulek, Elementary Algebraic Geometry, Student Mathematical Library, Vol. 20, AMS, Providence, RI, 2003. [Jacobson] N. Jacobson, Basic algebra I, Freeman and Co., San Francisco, 1974. [JK] T. Johnsen and S. Kleiman, Rational curves of degree at most 9 on a general quintic threefold, Comm. Algebra 24 (1996), 2721-2753. [Katz] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151-162. [Kontsevich] M. Kontsevich, Enumeration of rational curves via torus ac• tions, in The Moduli of Curves (Texel Island, 1994) (R. Dijk- graaf, C. Faber, and G. van der Geer, eds.), Progress in Math. 129, Birkhauser, Boston-Basel-Berlin, 1995, 335-368. [KM] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum co• homology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562. [Lang] S. Lang, Algebra, Addison-Wesley, New York, 1965. [LLY] B. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), 729-763. [MacLane] S. MacLane, Categories for the working mathematician, Grad• uate Texts in Mathematics 5, Springer-Verlag, New York, 1998. [MT] I. Madsen and J. Tornehave, From to cohomology: De Rham cohomology and characteristic classes, Cambridge University Press, Cambridge, 1997. [Manin] Yu. I. Manin, Gauge Field Theory and Complex Geometry, Springer-Verlag, Berlin, 1988. [MNOP] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donalds on-Thomas theory, I, preprint. [Miranda] R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics 5, AMS, Providence, RI, 1995. 200 Bibliography

[Munkres] J. R. Munkres, Topology: A first course, Prentice-Hall, Engle- wood Cliffs, NJ, 1975. [OV] H. Ooguri and C. Vafa, Knot Invariants and Topological Strings, Nucl. Phys. B577 (2000), 419-438. [Polchinski] J. Polchinski, String Theory, Vols. 1 and 2, Cambridge Uni• versity Press, Cambridge, 1998. [Reid] M. Reid, Undergraduate Algebraic Geometry, London Math. Society Student Texts 12, Cambridge University Press, 1988. [Shankar] R. Shankar, Principles of , Plenum Press, New York, 1994. [Spanier] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. [Spivak] M. Spivak, Calculus on Manifolds, Benjamin, Menlo Park, 1965. [Vick] J. W. Vick, Homology Theory: An Introduction to Algebraic Topol• ogy, 2nd ed., Grad. Texts in Math. 145, Springer-Verlag, New York, 1994. [Warner] F. W. Warner, Foundations of Differentiate Manifolds and Lie Groups, Grad. Texts in Math. 94, Springer-Verlag, New York, 1983. [Wells] R. O. Wells, Differential Analysis on Complex Manifolds, Grad. Texts in Math. 65, Springer-Verlag, New York, 1980. [Witten] E. Witten, Mirror Manifolds and Topological Field Theory, in Essays on Mirror Manifolds, S.-T. Yau (ed.), International Press, Hong Kong, 1992. Index

A-model, 163, 166, 176, 179, 180, 185 Bezout's , 16, 17, 19, 21, 22, action, 169, 172 24, 41, 80-81, 193, 194, see also correlation , 168-170, 189, enumerative geometry, in the see also Gromov-Witten plane invariant big quantum cohomology, 191-196 fermionic symmetry of, see BRST product, 194 symmetry associativity of, 194-196 observable, 169, 176 ofP1, 194-195 abelian group, 68 ofP2, 195-196 free, 70 bosonic string action, see string homomorphism of, 69, 76 theory, bosonic, action action, 136-137, 143, 146 boundary, 54, 72 bosonic string, see string theory, boundary operator, 71 bosonic, action branch point, 36, 38, 42 of A-model, see A-model, action BRST cohomology, 166, 168 supersymmetric, 150, 151 BRST operator, 166-170, 172 symmetry of, 139, 141 BRST symmetry, 165-170, see also continuous, 139 action, symmetry of, fermionic example of, 141 BRST-closed, 168 fermionic, 163 BRST-exact, 168 infinitesimal, 140, 143 , 31 CPn, see P71 and complex projective of, 32 space algebraic geometry, 9, 20 CP1, see P1 and complex projective angular momentum, 141 line antisymmetric product, 61, see also C°° , see manifold, C°° wedge product Calabi-Yau manifold, 187 area form, 160, 171 definition of, 174 genus, 34 example of, 175 atlas, 50, 58, 60 Calabi-Yau threefold, 132, 173-177 definition of, 174 example of, 174 B-model Gromov-Witten invariant of, 176 correlation function, 177, 189 , 137 ball, 44, 45 canonical bundle, 174, 184 Berezin integral, 149, 153, see also category, 48 fermionic integration example of, 49, 50, 76

201 202 Index

isomorphism in, 49, 55 conserved quantity, 139, 141 morphism in, 48, 49 continuous mapping, 47, 48 object in, 49 convergence, 46 Cauchy-Riemann , 163, 165 correlation function, 148, 178, 181, cellular decomposition, 81-83, 95-96, 183 99, 108 n-point, see n-point function chart, 50, see also atlas 0-point, see N^ and Np , 91, 105, 116, 130, 166 A-model, see A-model, correlation splitting principle, 106-108, 117, function 196 B-model, see B-model, correlation total, 106 function Clemens Conjecture, 29, 30, 131, 188 cotangent bundle closed set, 43-45, 55, see also holomorphic, 173 , closed of real manifold, 158 of cotangent space, 67 , 47 CW complex, 82 coboundary, 74, see also cohomology cycle cochain, 74, see also cohomology homology, 72 cocycle, 74, see also cohomology in tree of P^s, 32-34 cohomology, 40, 78, 85, 101 BRST, 166 de Rham cohomology, see cellular, 81, 82 cohomology, de Rham de Rham, 87, 166-176 deformation argument, 79 even, 78 degree group, 74 of hypersurface, 8 singular, 81, 82 of stable , 36 with real coefficients, 85 of subvariety of Pn, 85, 92 compactification, 5, 8, 21, 31 dehomogenization, see , complete conic, see conic, complete homogeneous, dehomogenization complex analysis, 59 derivation, 166 complex coordinates, 59 diffeomorphism, 58 complex manifold, 21, 59-60, 65, 173 , 67, 68, 85, 167 example of, 60 closed, 86 complex projective line, 4, see also exact, 86 P1 direct sum, 70 complex projective plane, 13, 20, see double cover, 36 also P2 dual basis, 100, 101 complex , 7, 55, 60, dual bundle, see line bundle, dual see also Pn dual conic, 39 manifold structure on, 52 dual plane, 17 complexification, 159, 161 conformal symmetry, 141 enumerative geometry, 1-11, 78 conic, 18-20, 25-26, 37 in the plane, 13-26 characteristic numbers of, 18, of motion, 139, 140, 143, 21-24, 40-41, 122-123 146, see also Euler-Lagrange complete, 24, 39 equation double line, 21, 23, 24, 38 equivalence relation, 10-11, 89, 102 dual, 39 quotient by, 4, 8, 11, 70 in quintic threefold, 30 Euclidean topology, 44, 47 line pair, 21, 34, 39 Euler's formula, 119 smooth, 20, 21, 38, 42 Euler-Lagrange equation, 138, 139 space of all, 21 evaluation map, 186 conservation law, 139, 141, see also excess bundle, 113-116, 121, 123 conserved quantity excess intersection, 24, 111-124, conservation of angular 130-131, 188 momentum, 141 expected dimension, see stable map, conservation of energy, 140, of, expected 143-144 dimension of conservation of momentum, 141 exterior , 86, 168, 172 Index 203

F0, 132, 171, 178, 194 homogenization, see polynomial, fermionic integration, 149, 153 homogenization fermionic symmetry, see action, homology, 80 symmetry of, fermionic cellular, 81, 82 field singular, 81, 82 bosonic, 148 homology class of a submanifold, 72, classical, 146 79 fermionic, 148 homology group, 72 quantum, 146-156 homomorphism theory, 145 image of, 69 flag, 99 kernel of, 69 Fourier series, 59 hyperplane, 9, see also hypersurface fundamental class, 79 at , 11 virtual, see virtual fundamental hyperplane bundle, 90 class hypersurface, 8, 14, 20, 42, 45 defining equation of, 8 degree of, 8 , 16, 19, 29, 105 singular, 20 genus 0 stable map, see stable map smooth, 20, 21 geometric vector bundle, see vector in projective space, 52, 54 bundle, geometric "good" enumerative problem, 13, 28, image, see homomorphism, image of 155, 187 theorem, 175 Gopakumar-Vafa invariant, 132 infinitesimal symmetry, see action, , 52, 95 symmetry of, infinitesimal cohomology of, 96-100, 108, 109 intersection multiplicity, 15, 16, universal bundle on, 102 21-23, 25, 39 Gromov-Witten invariant, 30, 128, intersection pairing, 78, 100, 190 130, 133, 166, 171, 176, 179, 188 intersection product, 73, 78, 79, 180 axioms intersection theory, 17, 18, 21 dimension axiom, 187, 189, 190 irreducible rational curve, see divisor axiom, 189, 193, 194 rational curve equivariance axiom, 187 definition of, 187 Jacobian, 64, 66 for Pn, 186, 188 example of, 189, 190 Gromov-Witten potential, 191-196, Kahler see also big quantum form, 169 cohomology manifold, 163, 164 of P1, 191-192 , 163 ofP2, 192-193 kernel, see homomorphism, kernel of group, see abelian group kinetic energy, 140

Lagrangian half-space, 44, 54 , 137, 140 Hermitian metric, 169 mechanics, 136-141 th Hilbert's 15 problem, 1, 13 , 46 Hodge theory, 133, 177 limit point, 46 holomorphic line, 9, 52-53 form, 174, 175, 178 in P3, see also Grassmannian function, 59 intersection of four, 97 mapping, 59, 165 in , 109 section of a line bundle, see line in quintic threefold, 30, 102, bundle, holomorphic section of 106-109, 178 section of a vector bundle, see in the plane, 17 vector bundle, holomorphic intersection of two, 2, 8, 11, 16, section of 18 homeomorphism, 50 line bundle, 88-93 homogeneous polynomial, see O(d), 90 polynomial, homogeneous direct sum, 106 204 Index

dual, 91 on manifold with boundary, 66 holomorphic section of, 90 oriented triangulation, 72 zero locus of, 90 isomorphism of, 89 product of, 91 p-form, 68, see also differential form trivialization of, 89 holomorphic, 173 line pair, 38 P™, 82, see also complex projective local coordinates, 50 space cohomology of, 83-84 M(Pn,d), see stable map, moduli hypersurface in, see hypersurface space of P1, see complex projective line

Mo,n(I, /?), see stable map, to big quantum cohomology of, see general X big quantum cohomology, 1 manifold, 50, 145 product, of P C°°, 57-59 P , see complex projective plane example of, 58-59 big quantum cohomology of, see atlas for, see atlas big quantum cohomology, compact oriented, 77 product, of P2 complex, see complex manifold quantum cohomology of, see example of, 51-54 quantum cohomology, of P2 orientable, 64—66 (P2)*, 17 manifold with boundary, 54-55, 76 pair of pants, 180, 182, 183 orientation on, see orientation, on manifold with boundary pairing meromorphic form, 175, see also between homology and holomorphic, form cohomology, 74 mirror manifold, 132 intersection, see intersection mirror symmetry, 30, 132, 176, 178, pairing 194 parametrization of a curve, 14, 19 mirror theorem, 30, 179, see also parametrized rational curve, 19, 20, quintic threefold 31, 37, 38, 41 moduli space of stable maps, see partition function, 147, 150, 155 stable map, moduli space of integral, 142, 144, 151, 165, 166, momentum, 140, 141 170, 189 morphism localization principle, 152-155, 165, from a tree of P^s to Pn, see tree 191 of P^s, morphism to P71 , 168 in a category, see also category, supersymmetric, 156 morphism in deformation invariance of, 155 multiple cover, see rational curve, in period, 178 quintic threefold, multiple cover Pfaffian, 156 of Pieri's formula, 101 Planck's constant h, 142 n-point function, 148, 189, 191, see plane curve, 14, 25 also correlation function enumerative geometry of, see also Nd, 130, 131, 170, 178, 194, see also enumerative geometry, in the Gromov-Witten invariant and plane Np Poincare , 75 N0, 170, 188 point at infinity, 4-5 Newton's law, 135-136, 139, 144 polynomial Noether's method, 140, 143 critical points of, 153-155 Noether's theorem, 139 homogeneous, 5, 19 normal bundle, 116, 117, 120-123 dehomogenization, 6, 88 homogenization, 6, 25 0(d), 90 root of in one variable, 2-3, 6, 155 open cover, 45 potential energy, 136, 140, 143 open set, 43, 44, see also topological potential function, 147 space, open subset of Gromov-Witten, see orientation, 60, 64, 73 Gromov-Witten potential Index 205 principle of conservation of number, spacetime, 145 13, 15, see also enumerative , 44, 66 geometry- splitting principle, see Chern class, probability amplitude, 142 splitting principle projective variety, 35 spring, 136, 140 irreducible, 35 stable map, 31, 36, 37, 40, 125, 130, projectivization, 7-8, 11, 52 131 pullback automorphism of, 36, 126-127, 133 of differential form, 87, 186 degree of, 36 of vector bundle, 128 isomorphism of, 36, 37 moduli space of, 37-39, 125, 166, quantum cohomology, 179, 183, 189, 185 see also topological quantum expected dimension of, 187 field theory, (1 + l)-dimensional to general X, 185-186 big, see big quantum cohomology stack, 37, 49, 127 of P2, 190 automorphism of, 129 product, 181, 189-190 standard n-simplex, 45, 65, 66, 71 associativity of, 182, 190, 196 orientation on, 66 quantum electrodynamics, 146 string theory, 1, 30-31, 141, 142, quantum field theory 157-172, 175 topological, see topological A-model, see A-model quantum field theory B-model, see B-model quantum gravity, 142 bosonic, 157, 163 quantum mechanics, 141, 147 action, 160, 163, 171 quintic mirror, 176-177, see also superstring, 163 mirror symmetry subgroup, 69, see also abelian group quintic threefold, 27, 132, 166, 170, submanifold, 53-54 173-175, 188, 189, 194 degree of, see degree, of subvariety as a Calabi-Yau threefold, 174 of Pn Gromov-Witten invariants of, see superstring theory, 163

Nd supersymmetry, 151, see also action, lines in, see line, in quintic supersymmetric threefold surface of genus g, 45, 55 quotient bundle, 116 symmetry quotient group, 70 conformal, 141 of action, see action, symmetry of rational curve, 19, 20, 25, 125, see also parametrized rational curve tangency, 15, 19, 21, 22, 41, 42 degree of, 19 tangent bundle, 117, 120 in quintic threefold, 27-31, holomorphic, 162 125-134, 171, 178 of Pn, 118-123 multiple cover of, 130-131 real, 171 real vector bundle, see vector bundle, tangent space, 63, 64 real topological gravity, 166, 181 residue, 175, 184 topological manifold, see manifold Riemann surface of genus g, see topological quantum field theory, 78, surface of genus g 179-183, 194 Riemannian metric, 158, 159, 163, (1 + l)-dimensional, 179-182, see 171 also quantum cohomology (2 -f l)-dimensional, 183 Schubert cycle, 95, 99-100, 107, 108, topological space, 43 see also Grassmannian closed subset of, 43 several complex variables, 161 compact, 5, 45, 46, 55 simplex, see standard n-simplex example of, 44-45, 55 singular chain, 71, see also homology, neighborhood of a point in, 43 singular open subset of, 43 singular point, 21 subspace of, 44 smooth point, 21 topological string amplitude, 128, see solutions at infinity, 5, 10, 45 also A-model, correlation 206 Index

function and Gromov-Witten invariant torsion, 74, 78, 100 TQFT, see topological quantum field theory transition function of a line bundle, 89 of a manifold, 51, 57, 60, 89 of a vector bundle, 102, 103, 115 transversal intersection, 73, 109, see also intersection theory tree ofPl!s, 31-36, 125 component of, 32 morphism to Pn, 35-36, 42, see also stable map node of, 32

variational derivative, 137-138 vector bundle, 102-109, 146 geometric, 103-105, 109 holomorphic section of, 103, 104, 109, 114 zero locus of, 105, 114-115, 130 isomorphism of, 102, 105 on a stack, 127, 129 real, 158 symmetric power of, 107 trivialization of, 102 virtual fundamental class, 186, 188

wedge product, 60 world-sheet, 30, 158 Titles in This Series

32 Sheldon Katz, Enumerative geometry and string theory, 2006 31 John McCleary, A first course in topology: Continuity and dimension, 2006 30 Serge Tabachnikov, Geometry and billiards, 2005 29 Kristopher Tapp, groups for undergraduates, 2005 28 Emmanuel Lesigne, Heads or tails: An introduction to limit in probability, 2005 27 Reinhard Illner, C. Sean Bohun, Samantha McCollum, and Thea van Roode, Mathematical modelling: A case studies approach, 2005 26 Robert Hardt, Editor, Six themes on variation, 2004 25 S. V. Duzhin and B. D. Chebotarevsky, Transformation groups for beginners, 2004 24 Bruce M. Landman and Aaron Robertson, Ramsey theory on the , 2004 23 S. K. Lando, Lectures on generating functions, 2003 22 Andreas Arvanitoyeorgos, An introduction to Lie groups and the geometry of homogeneous spaces, 2003 21 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis III: Integration, 2003 20 Klaus Hulek, Elementary algebraic geometry, 2003 19 A. Shen and N. K. Vereshchagin, Computable functions, 2003 18 V. V. Yaschenko, Editor, : An introduction, 2002 17 A. Shen and N. K. Vereshchagin, Basic , 2002 16 Wolfgang Kuhnel, Differential geometry: curves - surfaces - manifolds, second edition, 2006 15 Gerd Fischer, Plane algebraic curves, 2001 14 V. A. Vassiliev, Introduction to topology, 2001 13 Frederick J. Almgren, Jr., Plateau's problem: An invitation to varifold geometry, 2001 12 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis II: Continuity and differentiation, 2001 11 Michael Mesterton-Gibbons, An introduction to game-theoretic modelling, 2000 ® 10 John Oprea, The mathematics of soap films: Explorations with Maple , 2000 9 David E. Blair, Inversion theory and conformal mapping, 2000 8 Edward B. Burger, Exploring the number jungle: A journey into diophantine analysis, 2000 7 Judy L. Walker, Codes and curves, 2000 6 Gerald Tenenbaum and Michel Mendes France, The prime numbers and their distribution, 2000 TITLES IN THIS SERIES

5 Alexander Mehlmann, The game's afoot! in myth and paradox, 2000 4 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis I: Real numbers, and series, 2000 3 Roger Knobel, An introduction to the mathematical theory of , 2000 2 Gregory F. Lawler and Lester N. Coyle, Lectures on contemporary probability, 1999 1 Charles Radin, Miles of tiles, 1999