Mirror symmetry / Kentaro Hori ... [and others].

Format:

Book

Contributor:

Hori, Kentaro.

Rosengarten Family Fund.

Series:

Clay mathematics monographs 1539-6061 ; v. 1.

Clay mathematics monographs, 1539-6061 ; v. 1

Language:

English

Subjects (All):

Mirror symmetry.

Calabi-Yau manifolds.

Geometry, Enumerative.

Physical Description:

xx, 929 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society ; Cambridge, MA : Clay Mathematics Institute, [2003]

Contents:

A History of Mirror Symmetry xvii

Part 1. Mathematical Preliminaries 1

Chapter 1. Differential Geometry 3

1.2. Manifolds 4

1.3. Vector Bundles 5

1.4. Metrics, Connections, Curvature 11

1.5. Differential Forms 18

Chapter 2. Algebraic Geometry 25

2.2. Projective Spaces 25

2.3. Sheaves 32

2.4. Divisors and Line Bundles 38

Chapter 3. Differential and Algebraic Topology 41

3.2. Cohomology Theories 41

3.3. Poincare Duality and Intersections 42

3.4. Morse Theory 43

3.5. Characteristic Classes 45

Chapter 4. Equivariant Cohomology and Fixed-Point Theorems 57

4.1. A Brief Discussion of Fixed-Point Formulas 57

4.2. Classifying Spaces, Group Cohomology, and Equivariant Cohomology 58

4.3. The Atiyah-Bott Localization Formula 62

Chapter 5. Complex and Kahler Geometry 67

5.2. Complex Structure 67

5.3. Kahler Metrics 71

5.4. The Calabi-Yau Condition 74

Chapter 6. Calabi-Yau Manifolds and Their Moduli 77

6.2. Deformations of Complex Structure 79

6.3. Calabi-Yau Moduli Space 82

6.4. A Note on Rings and Frobenius Manifolds 87

6.5. Main Example: Mirror Symmetry for the Quintic 88

6.6. Singularities 95

Chapter 7. Toric Geometry for String Theory 101

7.2. Fans 102

7.3. GLSM 111

7.4. Intersection Numbers and Charges 114

7.5. Orbifolds 121

7.6. Blow-Up 123

7.7. Morphisms 126

7.8. Geometric Engineering 130

7.9. Polytopes 132

7.10. Mirror Symmetry 137

Part 2. Physics Preliminaries 143

Chapter 8. What Is a QFT? 145

8.1. Choice of a Manifold M 145

8.2. Choice of Objects on M and the Action S 146

8.3. Operator Formalism and Manifolds with Boundaries 146

8.4. Importance of Dimensionality 147

Chapter 9. QFT in d = 0 151

9.1. Multivariable Case 154

9.2. Fermions and Supersymmetry 155

9.3. Localization and Supersymmetry 157

9.4. Deformation Invariance 160

9.5. Explicit Evaluation of the Partition Function 162

9.6. Zero-Dimensional Landau-Ginzburg Theory 162

Chapter 10. QFT in Dimension 1: Quantum Mechanics 169

10.2. The Structure of Supersymmetric Quantum Mechanics 182

10.3. Perturbative Analysis: First Approach 197

10.4. Sigma Models 206

10.5. Instantons 220

Chapter 11. Free Quantum Field Theories in 1 + 1 Dimensions 237

11.1. Free Bosonic Scalar Field Theory 237

11.2. Sigma Model on Torus and T-duality 246

11.3. Free Dirac Fermion 254

Chapter 12. N = (2, 2) Supersymmetry 271

12.1. Superfield Formalism 271

12.3. N = (2, 2) Supersymmetric Quantum Field Theories 282

12.4. The Statement of Mirror Symmetry 284

Chapter 13. Non-linear Sigma Models and Landau-Ginzburg Models 291

13.2. R-Symmetries 294

13.3. Supersymmetric Ground States 299

13.4. Supersymmetric Sigma Model on T[superscript 2] and Mirror Symmetry 307

Chapter 14. Renormalization Group Flow 313

14.1. Scales 313

14.2. Renormalization of the Kahler Metric 315

14.3. Superspace Decouplings and Non-Renormalization of Superpotential 331

14.4. Infrared Fixed Points and Conformal Field Theories 335

Chapter 15. Linear Sigma Models 339

15.2. Supersymmetric Gauge Theories 348

15.3. Renormalization and Axial Anomaly 353

15.4. Non-Linear Sigma Models from Gauge Theories 356

15.5. Low Energy Dynamics 378

Chapter 16. Chiral Rings and Topological Field Theory 397

16.2. Twisting 399

16.3. Topological Correlation Functions and Chiral Rings 404

Chapter 17. Chiral Rings and the Geometry of the Vacuum Bundle 423

17.1. tt* Equations 423

Chapter 18. BPS Solitons in N =2 Landau-Ginzburg Theories 435

18.1. Vanishing Cycles 437

18.2. Picard-Lefschetz Monodromy 439

18.3. Non-compact n-Cycles 441

18.5. Relation Between tt* Geometry and BPS Solitons 447

Chapter 19. D-branes 449

19.2. Connections Supported on D-branes 452

19.3. D-branes, States and Periods 454

Part 3. Mirror Symmetry: Physics Proof 461

Chapter 20. Proof of Mirror Symmetry 463

20.2. Outline of the Proof 464

20.3. Step 1: T-Duality on a Charged Field 465

20.4. Step 2: The Mirror for Toric Varieties 472

20.5. Step 3: The Hypersurface Case 474

Part 4. Mirror Symmetry: Mathematics Proof 481

Chapter 22. Complex Curves (Non-singular and Nodal) 487

22.1. From Topological Surfaces to Riemann Surfaces 487

22.2. Nodal Curves 489

22.3. Differentials on Nodal Curves 491

Chapter 23. Moduli Spaces of Curves 493

23.1. Motivation: Projective Space as a Moduli Space 493

23.2. The Moduli Space M[subscript g] of Non-singular Riemann Surfaces 494

23.3. The Deligne-Mumford Compactification M[subscript g] of M[subscript g] 495

23.4. The Moduli Spaces M[subscript g,n] of Stable Pointed Curves 497

Chapter 24. Moduli Spaces M[subscript g,n](X, [beta]) of Stable Maps 501

24.1. Example: The Grassmannian 502

24.2. Example: The Complete (plane) Conics 502

24.3. Seven Properties of M[subscript g,n](X, [beta]) 503

24.4. Automorphisms, Deformations, Obstructions 504

Chapter 25. Cohomology Classes on M[subscript g,n] and M[subscript g,n](X, [beta]) 509

25.1. Classes Pulled Back from X 509

25.2. The Tautological Line Bundles L[subscript i], and the Classes [psi subscript i] 512

25.3. The Hodge Bundle E, and the Classes [lambda subscript kappa] 516

25.4. Other Classes Pulled Back from M[subscript g,n] 517

Chapter 26. The Virtual Fundamental Class, Gromov-Witten Invariants, and Descendant Invariants 519

26.1. The Virtual Fundamental Class 519

26.2. Gromov-Witten Invariants and Descendant Invariants 526

26.3. String, Dilaton, and Divisor Equations for M[subscript g,n](X, [beta]) 527

26.4. Descendant Invariants from Gromov-Witten Invariants in Genus 0 528

26.5. The Quantum Cohomology Ring 530

Chapter 27. Localization on the Moduli Space of Maps 535

27.1. The Equivariant Cohomology of Projective Space 535

27.2. Example: Branched Covers of P[superscript 1] 538

27.3. Determination of Fixed Loci 540

27.4. The Normal Bundle to a Fixed Locus 542

27.5. The Aspinwall-Morrison Formula 546

27.6. Virtual Localization 548

27.7. The Full Multiple Cover Formula for P[superscript 1] 551

Chapter 28. The Fundamental Solution of the Quantum Differential Equation 553

28.1. The "Small" Quantum Differential Equation 555

28.2. Example: Projective Space Revisited 556

Chapter 29. The Mirror Conjecture for Hypersurfaces I: The Fano Case 559

29.1. Overview of the Conjecture 559

29.2. The Correlators S(t, h) and S[subscript X](t, h) 562

29.3. The Torus Action 565

29.4. Localization 565

Chapter 30. The Mirror Conjecture for Hypersurfaces II: The Calabi-Yau Case 571

30.1. Correlator Recursions 571

30.2. Polynomiality 573

30.3. Correlators of Class P 577

30.4. Transformations 580

Part 5. Advanced Topics 583

Chapter 31. Topological Strings 585

31.1. Quantum Field Theory of Topological Strings 585

31.2. Holomorphic Anomaly 593

Chapter 32. Topological Strings and Target Space Physics 599

32.1. Aspects of Target Space Physics 599

32.2. Target Space Interpretation of Topological String Amplitudes 601

32.3. Counting of D-branes and Topological String Amplitudes 606

32.4. Black Hole Interpretation 612

Chapter 33. Mathematical Formulation of Gopakumar-Vafa Invariants 615

Chapter 34. Multiple Covers, Integrality, and Gopakumar-Vafa Invariants 635

34.1. The Gromov-Witten Theory of Threefolds 637

34.2. Proposal 639

34.3. Consequences for Algebraic Surfaces 641

34.4. Elliptic Rational Surfaces 643

34.5. Outlook 644

Chapter 35. Mirror Symmetry at Higher Genus 645

35.1. General Properties of the Genus 1 Topological Amplitude 645

35.2. The Topological Amplitude F[subscript 1] on the Torus 647

35.3. The Ray-Singer Torsion and the Holomorphic Anomaly 654

35.4. The Annulus Amplitude F[subscript ann] of the Open Topological String 657

35.5. F[subscript 1] on Calabi-Yau

in Three Complex Dimensions 662

35.6. Integration of the Higher Genus Holomorphic Anomaly Equations 668

35.7. Appendix A: Poisson Resummation 675

Chapter 36. Some Applications of Mirror Symmetry 677

36.1. Geometric Engineering of Gauge Theories 677

36.2. Topological Strings And Large N Chern-Simons Theory 680

Chapter 37. Aspects of Mirror Symmetry and D-branes 691

37.2. D-branes and Mirror Symmetry 692

37.3. D-branes in IIA and IIB String Theory 695

37.4. Mirror Symmetry as Generalized T-Duality 698

37.5. Mirror Symmetry with Bundles 704

37.6. Mathematical Characterization of D-branes 707

37.7. Kontsevich's Conjecture 709

37.8. The Elliptic Curve 714

37.9. A Geometric Functor 720

37.10. The Correspondence Principle 724

Chapter 38. More on the Mathematics of D-branes: Bundles, Derived Categories, and Lagrangians 729

38.2. Holomorphic Bundles and Gauge Theory 731

38.3. Derived categories 738

38.4. Lagrangians 744

Chapter 39. Boundary N = 2 Theories 765

39.1. Open Strings

Free Theories 766

39.2. Supersymmetric Boundary Conditions in N = 2 Theories 793

39.3. R-Anomaly 809

39.4. Supersymmetric Ground States 819

39.5. Boundary States and Overlap with RR Ground States 847

39.6. D-Brane Charge and Monodromy 859

39.7. D-Branes in N = 2 Minimal Models 870

39.8. Mirror Symmetry 884.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

0821829556

OCLC:

52374327