Topological Field Theory and Rational Curves

Abstract

We analyze the superstring propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear sigma-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.

20 pages

Cited by in corpus (109)

• Phases of $N=2$ Theories In Two Dimensions
• CFT's From Calabi-Yau Four-folds
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• Mirror Symmetry for Two Parameter Models -- II
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• Mirror Symmetry for Two Parameter Models -- I
• Calabi-Yau fourfolds for M- and F-Theory compactifications
• Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory
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• K3 Surfaces and String Duality
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• Gauge Theory at Large N and New G_2 Holonomy Metrics
• Mirror Manifolds in Higher Dimension
• Wall-Crossing from Boltzmann Black Hole Halos
• On the Ubiquity of K3 Fibrations in String Duality
• Counting higher genus curves in a Calabi-Yau manifold
• Considerations of One-Modulus Calabi-Yau Compactifications: Picard-Fuchs Equations, Kähler Potentials and Mirror Maps
• Measuring Small Distances in N=2 Sigma Models
• 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
• Hodge integrals and degenerate contributions
• Compactifications of moduli spaces inspired by mirror symmetry
• Multiple Mirror Manifolds and Topology Change in String Theory
• Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories
• Enumerative geometry of Calabi-Yau 4-folds
• Periods and Duality Symmetries in Calabi-Yau Compactifications
• Local Mirror Symmetry at Higher Genus
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• Elliptic Gromov - Witten invariants and the generalized mirror conjecture
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• Rational curves on hypersurfaces [after A. Givental]
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• Topological Sigma-Models in Four Dimensions and Triholomorphic Maps
• Differential geometry of the space of orbits of a Coxeter group
• Chiral Rings Do Not Suffice: N=(2,2) Theories with Nonzero Fundamental Group
• Recent Developments in (0,2) Mirror Symmetry
• A-twisted Landau-Ginzburg models
• The ice cone family and iterated integrals for Calabi-Yau varieties
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• Constraints For Topological Strings In $D\ge 1$
• Heterotic-Heterotic String Duality and Multiple K3 Fibrations
• The local Gromov-Witten theory of CP^1 and integrable hierarchies
• Exact Kahler Potential for Calabi-Yau Fourfolds
• Type B Topological Matter, Kodaira-Spencer Theory, and Mirror Symmetry
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• Les Houches Lectures on Fields, Strings and Duality
• Two-Sphere Partition Functions and Gromov-Witten Invariants
• The Coulomb Branch in Gauged Linear Sigma Models
• The Moduli Space of N = 2 Superconformal Field Theories
• Deformed Quantum Cohomology and (0,2) Mirror Symmetry
• Transversally Elliptic Complex and Cohomological Field Theory
• On Quantum Cohomology Rings for Hypersurfaces in $C{P}^{N-1}$
• Counting Curves with Modular Forms
• Rational Curves on the Space of Determinantal Nets of Conics
• Quantum K-Theory of Calabi-Yau Manifolds
• Quantum and Floer cohomology have the same ring structure
• String Duality and Enumeration of Curves by Jacobi Forms
• M-Theory, Topological Strings and Spinning Black Holes
• Differential Rings from Special Kähler Geometry
• Holomorphic Couplings In Non-Perturbative String Compactifications
• Gromov-Witten invariants and localization
• Recent Efforts in the Computation of String Couplings:
• Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint
• Open/closed Correspondence via Relative/local Correspondence
• Genus one enumerative invariants in P^n with fixed j invariant
• The local Gromov-Witten invariants of configurations of rational curves
• Effective computations of the Atiyah-Bott formula
• A Comparison Theorem for Gromov-Witten Invariants in the Symplectic Category
• Rational Curves on Calabi-Yau Threefolds
• Local B-model Yukawa couplings from A-twisted correlators
• Comments on Central Charge of Topological Sigma Model with Calabi-Yau Target Space
• A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Gromov-Witten Invariants via Algebraic Geometry
• Rational Curves and (0,2)-Deformations
• Genus-Zero Mirror Principle For Two Marked Points
• Sheaves of maximal intersection and multiplicities of stable log maps
• Special Geometry and Automorphic Forms
• Construction of Free Energy of Calabi-Yau manifold embedded in $C{P}^{n-1}$ via Torus Actions
• Notes on correlation functions in (0,2) theories
• Framing the Di-Logarithm (over Z)
• Constrained Topological Gravity from Twisted N=2 Liouville Theory
• On the multiplicity of reducible relative stable morphisms
• Determinantal Calabi-Yau varieties in Grassmannians and the Givental $I$-functions
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• A String Dual for Partially Topological Chern-Simons-Matter Theories
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• Open/Closed BPS Correspondence and Integrality
• Open Mirror Symmetry for Higher Dimensional Calabi-Yau Hypersurfaces
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• Computation of Yukawa Couplings for Calabi-Yau Hypersurfaces in Weighted Projective Spaces
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• IIA string instanton corrections to the four-fermion correlator in the intersection of Del Pezzo surfaces
• Embedded curves and Gromov-Witten invariants of three-folds
• Kähler Potential of Moduli Space of Calabi-Yau $d$-fold embedded in $C{P}^{d+1}$
• Quadratic Pseudostable Hodge Integrals and Mumford's Relations
• Gromov-Witten Theory and Threshold Corrections