Mirror Symmetry II
MirrorSymmetry II
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https://doi.org/10.1090/amsip/001AMS/IPStudies inAdvancedMathematicsVolume 1MirrorSymmetry IIB. Greene and S.-T . Yau, EditorsAmerican Mathematical Society International Press
Shing-l\ing Yau , Managin g Edito r2000 Mathematics Subject Classification. Primar y 1 4Nxx , 1 4Jxx , 32Gxx ,32J81, 81 T30 ,Library o f Congres s Cataloging-in-Publicatio n Dat aMirror symmetry II / B . Green e and S. T. Yau, editors.p. cm. — (AMS/IP studie s in advanced mathematics, ISSN 1 089-328 8 ; no. 1 )Includes bibliographical reference s and index.ISBN 0-8218-0634-3 (alk . paper)1. Mirror symmetry. 2 . Conformal invariants. 3 . Quantum field theory* 4 . Geometry, Enumerate. I . Greene, B. (Brian) , 1963- . II . Yau, Shing-Tung, 1 949 - , III . Series.QC174.17.S9M5631 99 6516.3'62—dc20 96-32938CIPCopying and reprinting. Materia l in this book may be reproduced by any means for educationaland scientific purpose s without fe e or permission with the exception of reproductio n b y service sthat collect fees for delivery of documents and provided that the customary acknowledgment of thesource is given. Thi s consent does not extend to other lands of copying for general distribution, foradvertising or promotional purposes , or for resale. Request s for permission for commercial use ofmaterial should be addressed to the Assistant to the Publisher, American Mathematical Society ,P. O* Box 6248, Providence , Rhod e Islan d 02940-6248. Request s can also b e made b y e-mail t oreprint permission«ams. org.Excluded from these provisions is material in articles for which the author holds copyright. I nsuch cases, requests for permission to use or reprint should be addressed directly to the author(s).(Copyright ownership is indicated in the notic e in the lower right-hand corner of the first pag e ofeach article.) c 1997 by the American Mathematical Societ y and International Press . Al l rights reserved.The American Mathematical Society and International Pres s retain all rightsexcept those granted to the United States Government.Printed in the United States of America.@ Th e paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.Visit th e AMS home page at URL : http://www.aias.org/10 9 8 7 6 5 4 3 07 06 05 04 03 02
ContentsI Constructio1n of Mirror ManifoldsGeometry and Quantum Field Theory: A Brief Introduction,by B.R. Greene, H. Ooguri 31 Introductio n 32 Quantu m Mechanics and the Path Integral 42.1 Bosonic Model 42.2 Supersymmetric Model 63 Th e Witten Index 94 Quantu m Mechanics on a1Riemannian Manifold14.1 Bosonic Model14.2 Supersymmetric Model5 Th e Supersymmetric Non-Linear Sigma Model and String Theory . . 16 Observable s and Ring Structures 27 Conclusion s 2224625Constructing Mirror Manifolds,by B.R. Greene 291 Introductio n 291.1 The Ingredients 302 Strateg y of the construction 333 Modul i Spaces of N 2 Superconformal Theories 364 Minima l Models and their automorphisms 404.1 Direct Calculation 435 Minima l Minimal Models and Calabi-Yau manifolds 455.1 Conjectured Correspondence 455.2 Arguments Establishing Minimal-Model/Calabi-Yau Correspondence 475.3 Generalizations546 Constructin g Mirror Manifolds 556.1 Orbifolding, Quantum Symmetries, and Unorbifolding . . 5 76.2 Details of the Calculation 596.3 Examples636.4 Implications657 Conclusion s 67Dual Cones and Mirror Symmetry,by Victor V. Batyrev, Lev A. Borisov 7 11 Introductio n 7 12 Reflexiv e Gorenstein cones 73v
vi CONTENT3 Fro m complete intersections to hypersurfaces 74 Complet e intersections and nef-partitions 85 Mirror s of rigid Calabi-Yau manifolds 8S602Mirror Symmetry Constructions: A Review,by Per Berglund, Sheldon Katz 871 Introductio n 872 N 2 Landau-Ginzburg models 892.1 Genera l framework 892.2 Fractiona l transformations 9 123 Transposition923 Th e toric approach 943.1 Tori c Generalities 943.2 Batyrev' s construction 963.3 Th e construction of Batyrev and Borisov 983.4 Transpositio1n via toric geometry 003.5 Example s 024 Discussion s 08On the Elliptic Genus and Mirror Symmetry,by Per11Berglund, M&ns Henningson1 Introductio n2 Landau-Ginzbur g orbifolds and11the elliptic genus112.1 Th e elliptic genus12.21Th e Poincar polynomial3 Mirro r symmetry for Landau-Ginzburg1orbifolds 213.1 Genera l results 23.2 Example s 255779003Orbifold Euler Characteristic,byShi-shyrRoan 291 Introductio n 2912 Loo p space of an orbifold 3 13 Equivarian1t K-theory interpretation 3214 Kleinia n Surface Singularities 325 Tori c resolution 336 Minima1l resolution of a double cover 37II Th 1e Structure of Moduli Space 4 1Phases of AT 2 Theories in Two Dimensions,by Edward Witten 43
CONTENTSvu1 Introductio n 4312 Fiel d Theory Background 4513 Th e C-Y/L-G Correspondence 543.1 The Model 5513.2 The "Singularity" 5713.3 Applications 643.4 Th 1e Twisted Models In Detail 6814 Som e Geometrical Background 755 Generalization s 795.1 Hypersurfaces In Weighted Projective1Space 795.2 Hypersurface1s In Toric Varieties 8 15.3 Hypersurface1s in Grassmannians 865.4 Intersection1s Of Hypersurfaces 8815.5 Change Of Topology 9216 Extensio n To (0,2) Models 9816.1 (0,2) Superfields 986.2 Some Models 203Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Changein String Theory,by Paul S. Aspinwall, Brian R. Greene, David R. Morrison1231 Introductio n 242 Mirro r Manifolds, Moduli Spaces and Topology1Change 2912.1 Mirror Manifolds 292.2 Conformal Field Theory Moduli Space 22 12.3 Complex Structure Moduli Space 2232.4 Kahler Structure Moduli Space 2242.5 Topology Change 2273 A Primer on Toric Geometry 23 13.1 Intuitive Ideas 23 13.2 The M an d N Lattice s 2333.3 Singularities and their Resolution 2373.4 Compactness and Intersections 24 13.5 Hypersurfaces in Toric Varieties 2423.6 KShler and Complex Structure Moduli 2453.7 Holomorphic Quotients 2453.8 Toric Geometry of the Partially Enlarged Kahler Moduli Space2473.9 Toric Geometry of the Complex Structure Moduli Space . 25 14 Mirro r Manifolds and Toric Geometry 2524.1 Toric Approach to Mirror Manifolds 2524.2 Complex Structure vs. Kahler Moduli Space: A Puzzle . . 25 3
viiiCONTENTS4.3 Asymptoti c Mirror Symmetry and The Monomial-DivisorMirror Map 255 A n Example 255.1 AMirror Pair of Calabi-Yau Spaces 255.2 Th e Moduli Spaces 265.3 Result s 265.4 Discussio n 266 Th e Fully Enlarged Kahler Moduli Space 267 Conclusion s 2756703665Picard-Fuchs Equations, Special Geometry and Target Space Duality,by A. Ceresole, R. D'Auria, S. Ferrara, W. Lerche, J. Louis and T. Regge2811 Introductio n and Summary 28 12 Differentia l equations for one variable 2882.1 Linea r differential equation s and W-generators 2882.2 Firs t order equations 2923 Differentia l equation s for arbitrary many moduli 2943.1 Holomorphi c Picard-Fuchs equations and special geometry 29 43.2 Non-holomorphi c Picard-Fuchs equations 3003.3 Singula r Picard-Fuchs systems 30 14 Relatio n to Calabi-Yau manifolds and topological fieldtheory . . . . 30 35 Targe t space duality and monodromy properties of Picard-Fuchs equations 3085.1 A n abelian subgroup of the duality group 3206 Dualit y group of an example with two moduli 3236.1 Introductio n 3236.2 Th e fundamental group of W(y\ a, b) 3256.3 Behaviou r of the periods around the singular curve . . . . 33 06.4 Th e monodromy generators 3346.5 Th e duality group 338Appendix A. Special Geometry 340A.l Kahler-Hodge manifolds 340A.2 Special Kahler manifolds 342Appendix B . Remarks on w3 0 and covariantly constant w4 345Appendix C . Differential equation s for cubic F-functions 347Resolution of Orbifold Singularities in String Theory,by Paul S- Aspinwall 351 Introductio n 352 Classica l Geometry 353 Conforma l Field Theory 364 Explorin g the Moduli Space 365 Quotien t Singularities in Two Dimensions 37559695
CONTENTSixThe Role of c2 in Calabi-Yau Classification - a Preliminary Survey,by R M.H.Wilson 38 11 Birationa l contractions and the KShler cone 3822 Fibr e space structures arising fromc2 3863 Th e case when c2 is strictly positive 389Thickening Calabi-Yau moduli spaces,by Z. Ran 391 Th e Schouten Lie algebra 392 Exoti c deformations 393 H 1 an d local systems 393458The Deformation Space of Calabi-Yau n-folds with Canonical SingularitiesCan Be Obstructed,by Mark Gross 40 11 Tw o Families of Calabi-Yau n-folds 4022 Th e Example 407Introduction to Duality,1by Amit Giveon, Martin Ro ek 411 Introductio n and summary 42 Sigma-model s 43 Isometrie s and1the duality transformation 44 Th e dual action 45 Quantu1m duality and global issues 45.1 Integration over the Lagrange1multiplier A 45.2 Integration1over the connection A 46 A n example: Abelian duality 426.1 The dual action 426.2 Fiel d Equations 426.3 Furthe r structures 42334568990023Non-compact Calabi-Yau Spaces and other Non-Trivial Backgrounds forFour-dimensional Superstrings,by E. Kiritsis, C. Kounnas, D. Lust 4271 Introductio n 4272 Th e AT 2 (AT 4) Background and U(l) Dualit y Transformations . . . 42 93 KShle r Spaces without Torsion and their Duals 43 14 Four-dimensiona l Non-Kahlerian Spaces with Torsion and their Duals 43 55 Conclusion s 439
x CONTENTSScaling Behavior On The Space Of Calabi-Yau Manifolds,by R. Schimmrigk 441 Introductio n 442 Th e Variables 443 Th e Class 444 Th e Results 4433467III Enumerativ e Issues and Mirror Symmetry 455Making enumerative predictions, b y Means of Mirro r Symmetryby David R. Morrison 4571 Coordinate s on the B-model moduli space . . . 4592 Th e large radius limit 46 12.1 Th e nonlinear a-model 46 12.2 Th e A-model parameter space 4632.3 Fla t coordinates and the large radius limit 4643 Maximall y unipotent monodromy 4664 Equivalenc e among boundary points 4705 Determinin g the mirror map (two conjectures) 4735.1 Aconjecture about integral cohomology 4735.2 The monomial-divisor mirror map 4756 Makin g enumerative predictions 477Mirror Symmetry for Two Parameter Models - 1,by Philip Candelas, Xenia de la Ossa, Anamaria Font, Sheldon Katz,David R. Morrison 481 Introductio n 482 Geometr y of Calabi-Yau Hypersurfaces in p(M,2,2,2) a nd p(i,i,2,2,6 ) 4 82.1 Linea r systems 482.2 Curves and the Kahler cone 482.3 Cher n classes 483 Th e Moduli Space of the Mirror 483.1 Basic facts 483.2 The locus f 2 1 493.3 More about the moduli space 494 Monodrom y and the Large Complex Structure Limit 494.1 The large complex structure limit 494.2 Monodromy calculations 495 Consideration s of Toric Geometry 506 Th e Periods 506.1 The fundamental period 506.2 The Picard-Fuchs equations 5033557899236682668
CONTENTS x6.3 Analytic properties of the fundamental period 506.4 Analytic continuation1of the periods 57 Th e Mirror Map and Large Complex Structure1Limit 57.1 Generalities57.2 The large complex structure limit for Piia,2,2,2) [8] 527.3 Inversion of the mirror map 528 Th e Yukawa Couplings and the Instanton Expansion 528.1 The couplings 528.2 Instantons of genus one 539 Verificatio n of Some Instanton Contributions 53i8599037724Mirror Symmetry, Mirror Map and Applications to Complete IntersectionCalabi-Yau Spaces,by S. Hosono, A. Klemm, S.Theisen and S.-T. Yau 5451 Introductio n 5452 Calculatio n of the classical topological data of CICYs 5483 Derivatio n of the Picard-Fuchs equations 5494 Loca l behaviour of the solutions, mirror map and instanton-correctedYukawa couplings 5555 Selecte d examples 5646 Connectio n with rational superconformal theories 5837 Topologica l one-loop partition function and the number of elliptic curves5878 Discussio n 595Appendix A. The pole structure in the coefficients of the logarithmic solutions to the Picard-Fuchs equation 596Appendix B . Predicted numbers of lines for complete intersections inP 3 x P 3 and P3 x P 2 60 1Gromov-Witten Classes, Quantum Cohomology,and Enumerative Geometry,by M. Kontsevich, Yu. Manin 6071 Introductio n 6072 Gromov-Witte n classes 60913 Firs t Reconstruction Theorem 694 Potential , associativity relations, and quantum cohomology 6225 Example s 6336 Cohomologica l Field Theory 6407 Homolog y of moduli spaces 6458 Secon d Reconstruction Theorem 648Holomorphic Anomalies in Topological Field Theories,by M. Bershadsky, S. Cecotti, H. Ooguri an d C. Vafa 651 Introductio n 6555
XllCONTENTS2 Topologica l Limit 653 Lo w Dimensional Examples and Quantum Mirror Symmetry 66Appendix A. Contact Term Contribution 67Appendix B . Intersection Theory over Moduli Spaces of DegenerateInstantons, by Sheldon Katz 67Local behavior of Hodge structures at infinity,by P. Deligne 6895243IV Mirro r Symmetry in Higher and Lower Dimensions 70 1String Theory on K3 Surfaces,by Paul S. As pin wall, David R. Morrison 701 Introductio n 702 Th e Moduli Space 7031Th e Space of Total Cohomology 74 Mirro r Symmetry for1Algebraic K3 Surfaces 733502K3 Surfaces with Involution and Mirror Pairs of Calabi-Yau Manifolds,by Ciprian Borcea 771 Th e mirror scenario 792 Calabi-Ya u manifolds with involution. The basic construction . . . . 72 03 Nikulin' s classification of K3 surfaces with involution 72 14 Mirro r pairs of Calabi-Yau threefolds 7235 Relation s with hypersurfaces in weighted projective spaces 7246 Relation s with Arnold's "strange duality" 7287 Reflexiv e polyhedra and Batyrev's duality 7308 Relation s with fibreproducts of rational elliptic surfaces with section 73 49 Othe r mirror pairs. Hilbert polynomials 73710 Highe r dimensions 739Mirror Manifolds in Higher Dimension,by Brian R. Greene, David R. Morrison and M. Ronen Plesser 7451 Introductio n 7452 Calabi-Ya u Moduli Spaces ford 3 . . . . . . . . . . 7482.1 Mathematical Preliminaries 7492.2 Picard-Fuchs Equations 75 12.3 Analog s of Special Geometry 7543 Yukaw a Couplings, Series Expansions and Factorization 7563.1 The Computation 7583.2 Factorization and Three-Point Functions 7604 Th e Mirror Map and Three-Point Functions 764
CONTENTS xiii4.1 The Gauss-Manin Connection and the Choice of Basis . . 75 74.2 Holomorphic Picard-Fuchs Equation and Three-Point Functions 7734.3 Factorization and the Other Yukawa Couplings 7765 Mathematica l Interpretatio n and Comparison of Instanton Sums . . . 77 96 Conclusion s 782Appendix A . Some Remarks on Covariant Derivatives 783Appendix B. The Multiple Cover Formula in Higher Dimension 784Supermanifolds, Rigid Manifolds and Mirror Symmetry,by S. Sethi 7931 Introductio n 7932 Landau-Ginzbur g Orbifolds an d Non-linear Sigma Models 7952.1 APath-Integral Argument 7952.2 Relation to Rigid Manifolds 7973 Supermanifold s 7983.1 Conditions for Conformal Invarianc e 7983.2 Super-Ricci Flat Metrics 8004 Th e Chiral Ring 8024.1 General Features 8024.2 The Hodge Structure 80514.3 Kahler Moduli 8 15 Conclusion s 82Glossary 86Index 839
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ForewordMirror symmetry has undergone dramatic progress since the Mathematical Sciences Research Institute workshop in 1 99 1 whose proceedings constitute volumeI of this continuing collection. Tremendou s insight has been gained on a numberof key issues, and it is the purpose of the present volume to survey some of theseresults. Some of the contributions are reprints of papers which have appeared elsewhere while others were written specifically for this collection.The areas covered are organized into four sections, and each presents papers byboth physicists and mathematicians. Section I focuses on the present understandingof explicit constructions of mirror manifolds. Paper 1 briefly reviews the notion ofpath integration to assist those less familiar with this physical tool. Paper 2 reviewsthe first,and at present, only known construction of mirror manifolds, at the levelof conformal field theory. Paper 3 discusses a more general construction of mirrorpairs, which as yet has not been established in conformal field theory, and paper 4reviews this and other conjectured constructions. Paper 5 discusses mirror symmetry in the context of Landau-Ginsburg theories and paper 6 reviews properties of theorbifolding operation, from a mathematical perspective.Section II focuses on work that has honed our understanding of both CalabiYau and conformal field theory moduli spaces. Paper s 7 and 8 discuss propertiesof the enlarged Kahler moduli space required by conformal field theory, and in particular, establish the firstconcrete arena for physically smooth spacetime topologychange. Pape r 9 discusses aspects of geometrical structur e of such moduli spacesand, in paper 10, some of the newfound understanding of moduli space is appliedto the case of orbifold theories. I n paper 11, the classification problem for CalabiYau's is discussed; in paper 1 2 Witten's notion of thickening the moduli space isdescribed from a mathematical perspective and in paper 1 3 an example of an obstructed moduli space is discussed. Paper 1 4 presents an introductory discussion ofduality properties of moduli space, paper 1 5 embarks on the issue of non-compactCalabi-Yau spaces while paper 1 6 discusses some interesting Calabi-Yau numerology.Section III focuses on developments in using mirror symmetry to solve difficultcounting problems, i.e. problem s in enumerative geometry. Paper s 1 8 and 1 9 discuss the methods for counting rational curves for examples whose parameter spaceis larger than one, while paper 1 7 presents a review of the multi-parameter case ingeneral. Pape r 20 places the physical approach to these enumerative problems onmore firmmathematical foundation, as well as applying such methods to a varietyof counting problems. Paper 21 resolves a number of key issues in mirror symmetrysuch as the form of the mirror map, in addition to providing a means of extending thedomain of accessible counting problems to higher genus curves. In paper 22, someaspects of the methods used in applying mirror symmetry to enumerative problemsxv
XVIFOREWORDare placed in an appropriate mathematical framework.Section IV focuses on the extension of mirror symmetry away from the familiarcase of complex dimension three to both lower and higher dimension. Papers 23 and24 discuss mirror symmetry for complex dimension 2 and papers 24 and 25 discussvarious aspects of mirror symmetry in dimension greater than 3.Due to space limitations, there are a number of equally interesting and important developments that have not been included. The papers of this volume, though,will undoubtedly allow the reader to gain much insight into both the physics and themathematics of the remarkable structure of mirror symmetry.The editors wish to thank Arthur Greenspoon and Misha Verbitsky, whose tremendous work and dedication has greatly improved the technical quality of this volume.
GlossaryP-Riemann symbol, 322ansatz, 434,587,752,758,765,78 9anti-chiral components- o f the gaugino field, 347anti-chiral fields, 651,652,660-primary, 51 ,74 0anti-chiral multiplets, 1 54,1 5 7anti-chiral operators- primary , 785anti-chiral perturbations, 652anti-DeRham algebra, 396anti-instanton field- o f instanton number (-k/n),1 7 9anti-instantons, 179,180Arnold's strange duality, 703,708,712,723,729,737- extensio n to Kodaira singularities, 724axion field, 428- antisymmetric , 291axionic instantons, 429,437Barnes type integral, 515Batyrev's duality, 725Beltrami differential, 66 8Berezin integration, 796, 802Bertin-Markushevich criterion for resolutions, 144Bianchi identities, 157,41 9- o n supercurvature, 347Bismut measure, twisted, 138Bogomolov-TIan-Todorovtheorem, 283,404,459,460BRST, 1 75 , 176, 180, 557, 651, 658,660, 795Calabi-Yau- background s (of the superconformal field theories), 428,429816- compactifications , 292,296,429,553,709- complet e intersections, 81, 85,87,88,90,91,94,107,113,114- , - line s on, 11 9- conforma l field theories, 64,219- doubl e covering, 733- ellipti c fibre spaces, 391- famil y- , mirro r of, 73 1- Ferma t hypersurfaces, 741-models, 1 62,1 65,1 69,1 7 1- , - conforma l invarianc e of ,152- orbifol d-,-definition, 36 0- , - a t the edge of the Kah
Geometry and Quantum Field Theory: A Brief Introduction, by B.R. Greene, H. Ooguri 3 1 Introduction 3 2 Quantum Mechanics and the Path Integral 4 2.1 Bosonic Model 4 2.2 Supersymmetric Model 6 3 The Witten Index 9 4 Quantum Mechanics on a Riemannian Manifold 12 4.1 Bosonic Model 12 4.2 Supersymmetric Model 14
Symmetry Point Groups Symmetry of a molecule located on symmetry axes, cut by planes of symmetry, or centered at an inversion center is known as point symmetry . Collections of symmetry operations constitute mathematical groups . Each symmetry point group has a particular designation. Cn, C nh, C nv Dn, D nh, D nd S2n C v ,D h
1/2 13: Symmetry 13.1 Line symmetry and rotational symmetry 4 To recognise shapes that have reflective symmetry To draw lines of symmetry on a shape To recognise shapes that have rotational symmetry To find the order of rotational symmetry for a shape 13.2 Reflections To
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mirror pair than the first type. For λ C, we say that Wλ is a strong mirror of Xλ. For such a strong mirror pair {Xλ,Wλ}, we can really ask for the relation between the zeta function of Xλ and the zeta function of Wλ. If λ1 6 λ2, Wλ 1 would not be called a strong mirror for Xλ 2, although they would be an usual weak mirror pair.
Chemistry of the following metals: (a) Li, Be, Ra (b) Sn, Pb. 4. Chemistry of halogens with reference of extraction, oxidation states and halides. GROUP-C: MISCELLANEOUS TOPICS 1. (a) Molecular Symmetry: An Introduction: Symmetry elements and symmetry operations, centre of symmetry, axis of symmetry and plane of symmetry (definitions). (b) Elementary Magnetochemistry: Types of magnetic .
Computational Mechanics, AAU, Esbjerg ANSYS Symmetry One of the most powerful means of reducing the size of a FEA problem is the exploitation of symmetry Symmetry is said to exist if there is a complete symmetry of geometry, loads and constraints about a line or plane of symmetry When exploiting symmetry model needs to be
SEMESTER I Paper I Molecular Symmetry and Molecular Vibrations 1. Molecular Symmetry: a) Symmetry elements and symmetry operations with special reference to water, ammonia and ethane. b) Classification of molecules/ ions based on their symmetry properties. c) Derivation of matrices for rotation, reflection, rotation-reflection and .
only line symmetry only rotational symmetry both line symmetry and rotational symmetry neither line symmetry and nor rotational symmetry 30 . These printable Worksheets and Practice Papers are available for FREE download Helps stu
