Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Abstract

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

59 pages. Some changes in the references, a few minor points have been clarified

References in corpus (14)

• Phases of $N=2$ Theories In Two Dimensions
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Holomorphic Anomalies in Topological Field Theories
• Mirror Symmetry for Two Parameter Models -- II
• A Generalized Construction of Mirror Manifolds
• Quantum Cohomology Rings of Toric Manifolds
• Considerations of One-Modulus Calabi-Yau Compactifications: Picard-Fuchs Equations, Kähler Potentials and Mirror Maps
• Compactifications of moduli spaces inspired by mirror symmetry
• Multiple Mirror Manifolds and Topology Change in String Theory
• Periods and Duality Symmetries in Calabi-Yau Compactifications
• Landau-Ginzburg String Vacua
• Generalized Calabi-Yau Manifolds and the Mirror of a Rigid Manifold
• Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space
• All Abelian Symmetries of Landau-Ginzburg Potentials

Cited by in corpus (266)

• Geometric Engineering of Quantum Field Theories
• Exact Results for N=2 Compactifications of Heterotic Strings
• Mirror Symmetry for Two Parameter Models -- II
• Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
• Nonperturbative Results on the Point Particle Limit of N=2 Heterotic String Compactifications
• K3--Fibrations and Heterotic-Type II String Duality
• On the Vacuum Structure of Type II String Compactifications on Calabi-Yau Spaces with H-Fluxes
• Calabi-Yau fourfolds for M- and F-Theory compactifications
• Tensionless Strings and the Weak Gravity Conjecture
• Evidence for Heterotic/Heterotic Duality
• Black Holes and Solitons in String Theory
• Enhanced Gauge Symmetry in Type II String Theory
• Branes, Calabi-Yau Spaces, and Toroidal Compactification of the N=1 Six-Dimensional E_8 Theory
• BPS States of Exceptional Non-Critical Strings
• A Stringy Test of the Scalar Weak Gravity Conjecture
• Flux Compactifications on Calabi-Yau Threefolds
• Classical and quantum N=2 supersymmetric black holes
• The Weak Gravity Conjecture: A Review
• On the Ubiquity of K3 Fibrations in String Duality
• Topological String Partition Functions as Polynomials
• String loop corrected hypermultiplet moduli spaces
• Counting higher genus curves in a Calabi-Yau manifold
• Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter
• N=2 Type II- Heterotic duality and Higher derivative F-terms
• N=2 Heterotic Superstring and its Dual Theory in Five Dimensions
• GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
• Arithmetic Properties of Mirror Map and Quantum Coupling
• Mirror Symmetry, N=1 Superpotentials and Tensionless Strings on Calabi-Yau Four-Folds
• Integrability of the holomorphic anomaly equations
• Axion Inflation in Type II String Theory
• On the Possibility of Large Axion Moduli Spaces
• The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces
• (0,2) Heterotic Gauge Couplings and their M-Theory Origin
• Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
• M-Theory on a Calabi-Yau Manifold
• Hypergeometric functions and mirror symmetry in toric varieties
• Strings of Minimal 6d SCFTs
• Aspects of Duality in N=2 String Vacua
• Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings
• Instanton sums and five-dimensional gauge theories
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Mirror Symmetry and Fano Manifolds
• On the Classification of Reflexive Polyhedra
• Small Cosmological Constants in String Theory
• Strong Coupling Singularities and Non-abelian Gauge Symmetries in $N=2$ String Theory
• Fibrations in CICY Threefolds
• Hints for Off-Shell Mirror Symmetry in type II/F-theory Compactifications
• Local Mirror Symmetry at Higher Genus
• Non-Perturbative Properties of Heterotic String Vacua Compactified on $K3\times T^2$
• Computing Brane and Flux Superpotentials in F-theory Compactifications
• Non-Perturbative Superpotentials in F-theory and String Duality
• Projections in string theory and boundary states for Gepner models
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• D-branes, Exceptional Sheaves and Quivers on Calabi-Yau manifolds: From Mukai to McKay
• On N=1 Mirror Symmetry for Open Type II Strings
• Resolution of Orbifold Singularities in String Theory
• Refined BPS invariants of 6d SCFTs from anomalies and modularity
• Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory
• Twistor Approach to String Compactifications: a Review
• New Higgs Transitions between Dual N=2 String Models
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs
• Summing the Instantons in Half-Twisted Linear Sigma Models
• D-branes on some one- and two-parameter Calabi-Yau hypersurfaces
• On Classifying the Divisor Involutions in Calabi-Yau Threefolds
• D3-instantons, Mock Theta Series and Twistors
• Conifold Vacua with Small Flux Superpotential
• Holomorphic Couplings in String Theory
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• EFT strings and emergence
• Exact Monodromy Group of N=2 Heterotic Superstring
• New Methods for Characterizing Phases of 2D Supersymmetric Gauge Theories
• Five-Brane Superpotentials and Heterotic/F-theory Duality
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• Perturbative Couplings and Modular Forms in N=2 String Models with a Wilson Line
• Landscaping with fluxes and the E8 Yukawa Point in F-theory
• Multiple D3-instantons and mock modular forms I
• Resolving Singularities in (0,2) Models
• Special Points of Inflation in Flux Compactifications
• Moduli Stabilization in Asymptotic Flux Compactifications
• Collapsing D-Branes in Calabi-Yau Moduli Space: I
• Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold
• Elliptic Blowup Equations for 6d SCFTs. II: Exceptional Cases
• Calculations for Mirror Symmetry with D-branes
• Blowup Equations for 6d SCFTs. I
• Type II/F-theory Superpotentials with Several Deformations and N=1 Mirror Symmetry
• Exact Kahler Potential for Calabi-Yau Fourfolds
• String Duality and Modular Forms
• Quantum periods of Calabi-Yau fourfolds
• Disc Instantons in Linear Sigma Models
• Instanton Numbers and Exchange Symmetries in $N=2$ Dual String Pairs
• Compactification, Geometry and Duality: N=2
• On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces
• String-dual $F_1$-Function in the Three Parameter Model
• A Search for Non-Perturbative Dualities of Local $N=2$ Yang--Mills Theories from Calabi--Yau Threefolds
• Separation of Attractors in 1-modulus Quantum Corrected Special Geometry
• On TCS $G_2$ manifolds and 4D Emergent Strings
• Systematics of type IIB moduli stabilisation with odd axions
• Mirror Symmetry for Toric Branes on Compact Hypersurfaces
• The rigid limit in Special Kahler geometry; From K3-fibrations to Special Riemann surfaces: a detailed case study
• Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries
• Type IIB flux vacua from G-theory II
• S-duality in Twistor Space
• (0,2) Deformations of Linear Sigma Models
• Multiple D3-instantons and mock modular forms II
• Some Navigation Rules for D-Brane Monodromy
• Dualities and fivebrane instantons
• Two-Sphere Partition Functions and Gromov-Witten Invariants
• Type IIB flux vacua from G-theory I
• Enhanced Symmetries in Multiparameter Flux Vacua
• Solitons in Seiberg-Witten Theory and D-branes in the Derived Category
• Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings
• Small flux superpotential in F-theory compactifications
• Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold
• Quantum N = 2 Supersymmetric Black Holes in the S-T Model
• A-Model Correlators from the Coulomb Branch
• Classical and Quantum Aspects of BPS Black Holes in N=2, D=4 Heterotic String Compactifications
• Moduli Space Reconstruction and Weak Gravity
• D(-1)-Instanton Superpotential In String Theory
• Local Mirror Symmetry: Calculations and Interpretations
• Quantum Lift of Non-BPS Flat Directions
• Rigid limit for hypermultiplets and five-dimensional gauge theories
• Lectures on Mirror Symmetry and Topological String Theory
• Mirror Maps, Modular Relations and Hypergeometric Series II
• Geometric engineering of (framed) BPS states
• The Moduli Space of N = 2 Superconformal Field Theories
• Subleading contributions from instanton corrections in N = 2 supersymmetric black hole entropy
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms
• Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds
• On Quantum Special Kaehler Geometry
• String Vacua with N=2 Supersymmetry in Four Dimensions
• Elliptic Blowup Equations for 6d SCFTs. IV: Matters
• Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions
• Perturbative gravitational couplings and Siegel modular forms in $D=4,N=2$ string models
• Heterotic Instanton Superpotentials from Complete Intersection Calabi-Yau Manifolds
• Deforming, revolving and resolving - New paths in the string theory landscape
• On Quantum Cohomology Rings for Hypersurfaces in $C{P}^{N-1}$
• Universal Properties of Type IIB and F-theory Flux Compactifications at Large Complex Structure
• Counting Curves with Modular Forms
• Making enumerative predictions by means of mirror symmetry
• Gopakumar-Vafa Hierarchies in Winding Inflation and Uplifts
• Quantum K-Theory of Calabi-Yau Manifolds
• A Note on the String Analog of $N=2$ Super-Symmetric Yang-Mills
• Quantum geometry of elliptic Calabi-Yau manifolds
• Superpotentials from Singular Divisors
• D-brane central charge and Landau-Ginzburg orbifolds
• Rigid Limit in N=2 Supergravity and Weak-Gravity Conjecture
• A Note on Computations of D-brane Superpotential
• Systematics of perturbatively flat flux vacua
• Topics in Cubic Special Geometry
• Special Geometry, Hessian Structures and Applications
• On quantum corrected Kähler potentials in F-theory
• Connecting Dualities and Machine Learning
• M-Theory, Topological Strings and Spinning Black Holes
• Exact solutions for some N=2 supersymmetric SO(N) gauge theories with vectors and spinors
• Open string mirror maps from Picard- Fuchs equations on relative cohomology
• Entropy Maximization in the Presence of Higher-Curvature Interactions
• Mirror Symmetry Constructions: A Review
• Duality Symmetries in N=2 Heterotic Superstring
• Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications
• Topological Strings on Non-Commutative Resolutions
• Heterotic Instantons for Monad and Extension Bundles
• Analytic Periods via Twisted Symmetric Squares
• Topological Strings on Elliptic Fibrations
• Moduli stabilization in type II Calabi-Yau compactifications at finite temperature
• Holomorphic Couplings In Non-Perturbative String Compactifications
• Matrix Factorizations, Massey Products and F-Terms for Two-Parameter Calabi-Yau Hypersurfaces
• Gromov-Witten invariants and localization
• A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear sigma model
• Emergent Strings in Type IIB Calabi--Yau Compactifications
• $K3$-Fibrations and Softly Broken $N=4$ Supersymmetric Gauge Theories
• Comments on M${}_{24}$ representations and $C{Y}_3$ geometries
• Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds
• The low-energy effective action for perturbative heterotic strings on $K_3\times T^2$ and the d=4 N=2 vector-tensor multiplet
• Prepotentials, Bi-linear Forms on Periods and Enhanced Gauge Symmetries in Type-II Strings
• Solving the Topological String on K3 Fibrations
• On the structure of supersymmetric T3 fibrations
• On genus-0 invariants of Calabi-Yau hybrid models
• The Ruled Vertex and D-E Degenerations
• Hypermultiplet metric and NS5-instantons
• Level Crossings, Attractor Points and Complex Multiplication
• Closed Sub-Monodromy Problems, Local Mirror Symmetry and Branes on Orbifolds
• Exact Solutions of Exceptional Gauge Theories from Toric Geometry
• Weyl invariant Jacobi forms along Higgsing trees
• Heterotic flux backgrounds and their IIA duals
• On genus one fibered Calabi-Yau threefolds with 5-sections
• The $\hat{\Gamma }$-genus and a regularization of an $S^1$-equivariant Euler class
• Type IIA on a Compact Calabi-Yau and D=11 Supergravity Uplift of its Orientifold
• Local B-model Yukawa couplings from A-twisted correlators
• Gromov-Witten Invariants via Algebraic Geometry
• $S$-Duality and the Calabi-Yau Interpretation of the $N=4$ to $N=2$ Flow
• Period Integrals and Tautological Systems
• The Master Equation for the Prepotential-Pub
• Enhanced Gauge Symmetry in Three-Moduli Models of Type-II String and Hypergeometric Series
• On Mathieu moonshine and Gromov-Witten invariants
• On (Orientifold of) type IIA on a Compact Calabi-Yau
• Special Geometry and Automorphic Forms
• N=1 Mirror Symmetry and Open/Closed String Duality
• Off-shell D-Brane/F-Theory Effective Superpotentials and Ooguri-Vafa Invariants of Several Compact Calabi-Yau Manifolds
• Towards $A+B$ theory in conifold transitions for Calabi-Yau threefolds
• Moduli mediation without moduli-induced gravitino problem
• Non-perturbative scalar potential inspired by type IIA strings on rigid CY
• ASK/PSK-correspondence and the r-map
• Determinantal Calabi-Yau varieties in Grassmannians and the Givental $I$-functions
• On D0-branes in Gepner models
• Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture
• On the mixed-twist construction and monodromy of associated Picard-Fuchs systems
• Mirror Symmetry of Calabi-Yau Manifolds and Flat Coordinates
• Instanton Corrected Non-Supersymmetric Attractors
• MQCD, ('Barely') G_2 Manifolds and (Orientifold of) a Compact Calabi-Yau
• Disk Amplitudes in Topological A-Model on Calabi-Yau Spaces and Fano Manifolds
• Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface
• Heterotic strings on $(K3\times T^2)/{\mathbb{Z}}_3$ and their dual Calabi-Yau threefolds
• Prepotentials from Symmetric Products
• Non Perturbative Check of N=2, D=4 Heterotic/Type II Duality
• On Semi-Periods
• Mirror Symmetry, D-brane Superpotential and Ooguri-Vafa Invariants of Compact Calabi-Yau Manifolds
• On the Stability of Non-Supersymmetric Quantum Attractors in String Theory
• Type II/F-theory Superpotentials and Ooguri-Vafa Invariants of Compact Calabi-Yau Threefolds with Three Deformations
• Local B-model and Mixed Hodge Structure
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants
• Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops
• Post Inflationary Behaviour of String Moduli
• Asymptotic curvature of moduli spaces for Calabi-Yau threefolds
• Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-Hübsch-Krawitz duality
• Geometric Engineering of Seiberg-Witten Theories with Massive Hypermultiplets
• Mirror Symmetry and Modularity
• The Weak Gravity Conjecture and BPS Particles
• Axion decay constants at special points in type II string theory
• Evaluation of Periods via Fibrations in Seiberg-Witten Theories and in Type-II String
• Open Mirror Symmetry for Higher Dimensional Calabi-Yau Hypersurfaces
• Reflections of NS5 branes
• Non-Supersymmetric Stringy Attractors
• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture
• B-type D-branes in hybrid models
• Conformal bootstrap and Mirror symmetry of states in Gepner models
• Noncommutative resolutions and CICY quotients from a non-abelian GLSM
• Toric Residues and Mirror Symmetry
• Differential equations satisfied by modular forms and K3 surfaces
• Topological String on elliptic CY 3-folds and the ring of Jacobi forms
• Effective Superpotentials of Type II D-brane/F-theory on Compact Complete Intersection Calabi-Yau Threefolds
• The Physical Mirror Equivalence for the Local P^2
• Ooguri-Vafa Invariants and Off-shell Superpotentials of Type II/F-theory compactification
• Strings from Membranes and Fivebranes
• Superpotentials of D-branes in Calabi-Yau Manifolds with Several Moduli by Mirror Symmetry and Blown-up
• GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities
• Dualities between K3 fibered Calabi-Yau three-folds
• Calabi--Yau Phases of the Superstring Vacuum
• D-brane Superpotentials and Ooguri-Vafa Invariants of Compact Calabi-Yau Threefolds
• Study of the OSV Conjecture for 4D N=2 Extremal Black Holes in Type IIB String Theory
• Tautological systems under the conifold transition on $G(2,4)$
• Ramanujan-type $1/\pi$-series from bimodular forms
• Mirror symmetry for double cover Calabi--Yau varieties
• Supergravity on R4 x S1/Z2 and singular Calabi-Yaus
• Local Mirror Symmetry and Type IIA Monodromy of Calabi-Yau manifolds
• Gromov-Witten Theory and Threshold Corrections
• The Number of Rational Points of Two Parameter Calabi-Yau manifolds as Toric Hypersurfaces
• On Stokes Matrices of Calabi-Yau hypersurfaces
• Mirror Symmetry of Calabi-Yau Manifolds Fibered by (1,8)-Polarized Abelian Surfaces
• Mirror Symmetry and the Web of Landau-Ginzburg String Vacua
• Calabi-Yau compactification of type II string theories
• Effective Actions Near Singularities: The STU-Model
• Direct Computation of Monodromy Matrices and Classification of 4d N=2 Heterotic--IIA Dual Vacua
• No inflation in type IIA strings on rigid CY spaces
• On ordinary crystals with logarithmic poles