Mirror Symmetry

Mirror Symmetry

by Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande
     
 

ISBN-10: 0821829556

ISBN-13: 9780821829554

Pub. Date: 08/19/2003

Publisher: American Mathematical Society

Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ''mirror'' geometry. The

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Overview

Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ''mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar Vafa invariants. This book aims to give a single, cohesive treatment of mirror symmetry from both the mathematical and physical viewpoint. Parts 1 and 2 develop the necessary mathematical and physical background ''from scratch,'' and are intended for readers trying to learn across disciplines. The treatment is focussed, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ''pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topics in mirror symmetry, including the role of D-branes in the context of mirror symmetry, and some of their applications in physics and mathematics: topological strings and large $N$ Chern-Simons theory; geometric engineering; mirror symmetry at higher genus; Gopakumar-Vafa invariants; and Kontsevich's formulation of the mirror phenomenon as an equivalence of categories. This book grew out of an intense, month-long course on mirror symmetry at Pine Manor College, sponsored by the Clay Mathematics Institute. The lecturers have tried to summarize this course in a coherent, unified text.

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Product Details

ISBN-13:
9780821829554
Publisher:
American Mathematical Society
Publication date:
08/19/2003
Series:
Clay Mathematics Monographs, #1
Edition description:
New Edition
Pages:
929
Product dimensions:
7.30(w) x 10.30(h) x 2.20(d)

Table of Contents

Preface
Introduction
Pt. 1Mathematical Preliminaries1
Ch. 1Differential Geometry3
Ch. 2Algebraic Geometry25
Ch. 3Differential and Algebraic Topology41
Ch. 4Equivariant Cohomology and Fixed-Point Theorems57
Ch. 5Complex and Kahler Geometry67
Ch. 6Calabi Yan Manifolds and Their Moduli77
Ch. 7Toric Geometry for String Theory101
Pt. 2Physics Preliminaries143
Ch. 8What Is a QFT?145
Ch. 9QFT in d = 0151
Ch. 10QFT in Dimension 1: Quantum Mechanics169
Ch. 11Free Quantum Field Theories in 1 + 1 Dimensions237
Ch. 12N = (2,2) Supersymmetry271
Ch. 13Non-linear Sigma Models and Landau-Ginzburg Models291
Ch. 14Renormalization Group Flow313
Ch. 15Linear Sigma Models339
Ch. 16Chiral Rings and Topological Field Theory397
Ch. 17Chiral Rings and the Geometry of the Vacuum Bundle423
Ch. 18BPS Solitons N = 2 Landau - Ginzburg Theories435
Ch. 19D-branes449
Pt. 3Mirror Symmetry: Physics Proof461
Ch. 20Proof of Mirror Symmetry463
Pt. 4Mirror Symmetry: Mathematics Proof481
Ch. 21Introduction and Overview483
Ch. 22Complex Curves (Non-singular and Nodal)487
Ch. 23Moduli Spaces of Curves493
Ch. 24Moduli Spaces [actual symbol not reproducible] of Stable Maps501
Ch. 25Cohomology Classes on [actual symbol not reproducible] and [actual symbol not reproducible]509
Ch. 26The Virtual Fundamental Class, Gromov-Witten Invariants, and Descendant Invariants519
Ch. 27Localization on the Moduli Space of Maps535
Ch. 28The Fundamental Solution of the Quantum Differential Equation553
Ch. 29The Mirror Conjecture for Hypersurfaces I: The Fano Case559
Ch. 30The Mirror Conjecture for Hypersurfaces II: The Calabi-Yau Case571
Pt. 5Advanced Topics583
Ch. 31Topological Strings585
Ch. 32Topological Strings and Target Space Physics599
Ch. 33Mathematical Formulation of Gopakumar-Vafa Invariants615
Ch. 34Multiple Covers, Integrality, and Gopakumar - Vafa Invariants635
Ch. 35Mirror Symmetry at Higher Genus645
Ch. 36Some Applications of Mirror Symmetry677
Ch. 37Aspects of Mirror Symmetry and D-branes691
Ch. 38More on the Mathematics of D-branes: Bundles, Derived Categories, and Lagrangians729
Ch. 39Boundary N = 2 Theories765
Ch. 40References889
Bibliography905
Index921

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