Mirror Symmetry

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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: 8/19/2003
  • 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

Pt. 1 Mathematical Preliminaries 1
Ch. 1 Differential Geometry 3
Ch. 2 Algebraic Geometry 25
Ch. 3 Differential and Algebraic Topology 41
Ch. 4 Equivariant Cohomology and Fixed-Point Theorems 57
Ch. 5 Complex and Kahler Geometry 67
Ch. 6 Calabi Yan Manifolds and Their Moduli 77
Ch. 7 Toric Geometry for String Theory 101
Pt. 2 Physics Preliminaries 143
Ch. 8 What Is a QFT? 145
Ch. 9 QFT in d = 0 151
Ch. 10 QFT in Dimension 1: Quantum Mechanics 169
Ch. 11 Free Quantum Field Theories in 1 + 1 Dimensions 237
Ch. 12 N = (2,2) Supersymmetry 271
Ch. 13 Non-linear Sigma Models and Landau-Ginzburg Models 291
Ch. 14 Renormalization Group Flow 313
Ch. 15 Linear Sigma Models 339
Ch. 16 Chiral Rings and Topological Field Theory 397
Ch. 17 Chiral Rings and the Geometry of the Vacuum Bundle 423
Ch. 18 BPS Solitons N = 2 Landau - Ginzburg Theories 435
Ch. 19 D-branes 449
Pt. 3 Mirror Symmetry: Physics Proof 461
Ch. 20 Proof of Mirror Symmetry 463
Pt. 4 Mirror Symmetry: Mathematics Proof 481
Ch. 21 Introduction and Overview 483
Ch. 22 Complex Curves (Non-singular and Nodal) 487
Ch. 23 Moduli Spaces of Curves 493
Ch. 24 Moduli Spaces [actual symbol not reproducible] of Stable Maps 501
Ch. 25 Cohomology Classes on [actual symbol not reproducible] and [actual symbol not reproducible] 509
Ch. 26 The Virtual Fundamental Class, Gromov-Witten Invariants, and Descendant Invariants 519
Ch. 27 Localization on the Moduli Space of Maps 535
Ch. 28 The Fundamental Solution of the Quantum Differential Equation 553
Ch. 29 The Mirror Conjecture for Hypersurfaces I: The Fano Case 559
Ch. 30 The Mirror Conjecture for Hypersurfaces II: The Calabi-Yau Case 571
Pt. 5 Advanced Topics 583
Ch. 31 Topological Strings 585
Ch. 32 Topological Strings and Target Space Physics 599
Ch. 33 Mathematical Formulation of Gopakumar-Vafa Invariants 615
Ch. 34 Multiple Covers, Integrality, and Gopakumar - Vafa Invariants 635
Ch. 35 Mirror Symmetry at Higher Genus 645
Ch. 36 Some Applications of Mirror Symmetry 677
Ch. 37 Aspects of Mirror Symmetry and D-branes 691
Ch. 38 More on the Mathematics of D-branes: Bundles, Derived Categories, and Lagrangians 729
Ch. 39 Boundary N = 2 Theories 765
Ch. 40 References 889
Bibliography 905
Index 921
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