The Topological Vertex

Abstract

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact Calabi-Yau toric threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of Calabi-Yau. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the Kodaira-Spencer quantum theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

70 pages, 16 figures, harvmac

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• Knot Invariants and Topological Strings
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Cited by in corpus (331)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Topological Strings and Integrable Hierarchies
• The Refined Topological Vertex
• Curve counting via stable pairs in the derived category
• Quantum Foam and Topological Strings
• Open string amplitudes and large order behavior in topological string theory
• Non-perturbative effects and the refined topological string
• The Holographic Entropy Cone
• Holomorphic Blocks in Three Dimensions
• Remodeling the B-model
• Matrix Models, Geometric Engineering and Elliptic Genera
• Torus knots and mirror symmetry
• Quantum Geometry of Refined Topological Strings
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Direct Integration of the Topological String
• Factorisation of N = 2 theories on the squashed 3-sphere
• Refined stable pair invariants for E-, M- and [p,q]-strings
• Refined, Motivic, and Quantum
• Refined Topological Vertex and Instanton Counting
• A & B model approaches to surface operators and Toda theories
• A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
• Brane Tilings and Their Applications
• Quantum Algebraic Approach to Refined Topological Vertex
• W-symmetry, topological vertex and affine Yangian
• Extended Holomorphic Anomaly in Gauge Theory
• Crystal Melting and Toric Calabi-Yau Manifolds
• On orbifolds of M-Strings
• Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
• Instanton counting, Macdonald function and the moduli space of D-branes
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Top Down Approach to 6D SCFTs
• Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Topological strings and 5d T_N partition functions
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• M5-branes, toric diagrams and gauge theory duality
• BPS Degeneracies and Superconformal Index in Diverse Dimensions
• Fermions from Half-BPS Supergravity
• A-polynomial, B-model, and Quantization
• Fiber-Base Duality and Global Symmetry Enhancement
• Ghost D-branes
• Seiberg-Witten theory as a Fermi gas
• Link Homologies and the Refined Topological Vertex
• The D5-brane effective action and superpotential in N=1 compactifications
• Vertices, Vortices & Interacting Surface Operators
• The ABCDEFG of Instantons and W-algebras
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
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• Refined BPS invariants of 6d SCFTs from anomalies and modularity
• Topological Strings and Quantum Spectral Problems
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• Seiberg-Witten theory and matrix models
• Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey
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• Phase transitions, double-scaling limit, and topological strings
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• Quivers as Calculators: Counting, Correlators and Riemann Surfaces
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
• Effective superpotentials for compact D5-brane Calabi-Yau geometries
• Topological strings and large N phase transitions II: Chiral expansion of q-deformed Yang-Mills theory
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Shifted Quiver Yangians and Representations from BPS Crystals
• Anomaly in RTT relation for DIM algebra and network matrix models
• On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
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• The Liouville side of the Vortex
• Instanton-Monopole Correspondence from M-Branes on ${\mathbb{S}}^1$ and Little String Theory
• Changing the preferred direction of the refined topological vertex
• Calculations for Mirror Symmetry with D-branes
• Rank-3 antisymmetric matter on 5-brane webs
• Quiver elliptic W-algebras
• The toroidal block and the genus expansion
• Wall Crossing, Quivers and Crystals
• The Geometry of D-Brane Superpotentials
• String theory and the Kauffman polynomial
• Counting BPS states on the Enriques Calabi-Yau
• Decomposing Nekrasov Decomposition
• Exact partition functions of Higgsed 5d $T_N$ theories
• Blowup Equations for 6d SCFTs. I
• A slow review of the AGT correspondence
• N=2 Supersymmetric Black Attractors in Six and Seven Dimensions
• Holomorphic blocks and the 5d AGT correspondence
• The local Gromov-Witten theory of CP^1 and integrable hierarchies
• The Wave Function Behavior of the Open Topological String Partition Function on the Conifold
• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Bootstrapping BPS spectra of 5d/6d field theories
• Complete prepotential for 5d $\mathcal{N}=1$ superconformal field theories
• Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds
• Crepant resolutions and open strings
• Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions
• A Solvable Deformation of Quantum Mechanics
• Geometric transitions and integrable systems
• Quantization condition from exact WKB for difference equations
• An elliptic Virasoro symmetry in 6d
• Topological strings, strips and quivers
• Quiver Matrix Model and Topological Partition Function in Six Dimensions
• Exact 2-point function in Hermitian matrix model
• D-branes as a Bubbling Calabi-Yau
• A universe field theory for JT gravity
• Self-Duality and Self-Similarity of Little String Orbifolds
• Topological strings and large N phase transitions I: Nonchiral expansion of q-deformed Yang-Mills theory
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• Topological open strings on orbifolds
• On elementary proof of AGT relations from six dimensions
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Open topological strings and integrable hierarchies: Remodeling the A-model
• Exact results for topological strings on resolved Y(p,q) singularities
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• Topological vertex formalism with O5-plane
• Quantum Entanglement of Baby Universes
• Wilson Loops, Geometric Transitions and Bubbling Calabi-Yau's
• Large N duality, lagrangian cycles, and algebraic knots
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Matrix models for 2* theories
• Dual Little Strings and their Partition Functions
• Crystal Melting and Wall Crossing Phenomena
• Localization at large N in Chern-Simons-matter theories
• Instanton Effects in Rank Deformed Superconformal Chern-Simons Theories from Topological Strings
• Macdonald operators and homological invariants of the colored Hopf link
• Wall Crossing As Seen By Matrix Models
• The Volume Conjecture and Topological Strings
• Crystal Model for the Closed Topological Vertex Geometry
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Calabi-Yau geometry and electrons on 2d lattices
• 6d strings and exceptional instantons
• Refined geometric transition and $qq$-characters
• Instantons from Blow-up
• Refined Black Hole Ensembles and Topological Strings
• Topological Strings and (Almost) Modular Forms
• Black-holes, topological strings and large N phase transitions
• Hopf link invariants and integrable hierarchies
• A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories
• Branes, Black Holes and Topological Strings on Toric Calabi-Yau Manifolds
• Exact Deconstruction of the 6D (2,0) Theory
• New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds
• Amoebas and Instantons
• On the Elliptic Genus of Three E-strings and Heterotic Strings
• From Little String Free Energies Towards Modular Graph Functions
• Flop invariance of the topological vertex
• Topological string amplitudes for the local half K3 surface
• Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials
• 3d Mirror Symmetry from S-duality
• Beyond Triality: Dual Quiver Gauge Theories and Little String Theories
• Quantum toroidal algebras and solvable structures in gauge/string theory
• The Real Topological Vertex at Work
• Stable maps to Looijenga pairs
• 5d/6d Wilson loops from blowups
• More on topological vertex formalism for 5-brane webs with O5-plane
• Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra
• The Quantum McKay Correspondence for polyhedral singularities
• Gromov-Witten theory of A_n-resolutions
• Knot invariants and Calabi-Yau crystals
• Instanton counting and O-vertex
• Crystal melting on toric surfaces
• Genus zero Gopakumar-Vafa invariants from open strings
• Fiber-base duality from the algebraic perspective
• Non-compact Topological Branes on Conifold
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Kerov functions revisited
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• Special colored Superpolynomials and their representation-dependence
• Refined Hopf Link Revisited
• Macdonald topological vertices and brane condensates
• 3d Expansions of 5d Instanton Partition Functions
• Black Holes in Type IIA String on Calabi-Yau Threefolds with Affine ADE Geometries and q-Deformed 2d Quiver Gauge Theories
• Five-Dimensional Gauge Theories from Shifted Web Diagrams
• Cauchy formula and the character ring
• Web Construction of ABCDEFG and Affine Quiver Gauge Theories
• Higgsing and Twisting of 6d $D_N$ gauge theories
• Plane Partition Realization of (Web of) W-algebra Minimal Models
• Cobordism invariants from BPS q-series
• Partition functions of web diagrams with an O7${}^{-}$-plane
• Five-Brane Webs and Highest Weight Representations
• Universal Character and Large N Factorization in Topological Gauge/String Theory
• Generalized MacMahon G(q) as q-deformed CFT Correlation Function
• Special geometry of local Calabi-Yau manifolds and superpotentials from holomorphic matrix models
• Introduction to the Gopakumar-Vafa Large N Duality
• A Macdonald refined topological vertex
• Topological vertex for 6d SCFTs with ${\mathbb{Z}}_2$-twist
• Entanglement entropy and edge modes in topological string theory II: The dual gauge theory story
• A Vertex Formalism for Local Ruled Surfaces
• Challenges of beta-deformation
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• The Gerby Gopakumar-Mariño-Vafa Formula
• An elliptic topological vertex
• The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories
• Dihedral Symmetries of Gauge Theories from Dual Calabi-Yau Threefolds
• Systematics of perturbatively flat flux vacua
• Spectral determinants and quantum theta functions
• A Note on Computations of D-brane Superpotential
• A reverse isoperimetric inequality for J-holomorphic curves
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Refined topological vertex with ON-planes
• Exact results for ABJ Wilson loops and open-closed duality
• Surface defects on E-string from 5-brane webs
• Playing with the index of M-theory
• A non-torus link from topological vertex
• Trinion Conformal Blocks from Topological strings
• Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane
• Can tangle calculus be applicable to hyperpolynomials?
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• 6d/5d exceptional gauge theories from web diagrams
• The remodeling conjecture and the Faber-Pandharipande formula
• Seiberg-Witten curves with O7${}^{\pm }$-planes
• A Nekrasov-Okounkov formula for Macdonald polynomials
• Gauge origami and quiver W-algebras
• Branes, quivers and wave-functions
• Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs
• Holomorphic Couplings In Non-Perturbative String Compactifications
• Surface Operators from M-strings
• A matrix model for the topological string II: The spectral curve and mirror geometry
• Topological Vertex/anti-Vertex and Supergroup Gauge Theory
• Refining the Shifted Topological Vertex
• A note on S-dual basis in free fermion system
• Crystal Melting, BPS Quivers and Plethystics
• Check-Operators and Quantum Spectral Curves
• Towards U(N|M) knot invariant from ABJM theory
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Blowup Equations for Little Strings
• Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds
• Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space
• 5D N=1 super QFT: symplectic quivers
• Refined topological vertex for a 5D $Sp(N)$ gauge theory with antisymmetric matter
• Open Gromov-Witten Theory of $K_{{\mathbb{P}}^2},K_{{\mathbb{P}}^1\times {\mathbb{P}}^1},K_{W\mathbb{P}[1,1,2]},K_{{\mathbb{F}}_1}$ and Jacobi Forms
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• Gromov-Witten Invariants of Local P^2 and Modular Forms
• Toric geometry of $G_2$-manifolds
• The Ruled Vertex and Nontoric del Pezzo Surfaces
• BPS States Meet Generalized Cohomology
• The Condensate from Torus Knots
• Refined Cigar and Omega-deformed Conifold
• Topological string entanglement
• The Ruled Vertex and D-E Degenerations
• Trace Identities for the Topological Vertex
• Open string amplitudes of closed topological vertex
• Worldsheet Interpretation of the Level-Rank Duality
• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• The local Gromov-Witten invariants of configurations of rational curves
• From topological strings to minimal models
• Modular properties of full 5D SYM partition function
• The Enhancement of Supersymmetry in M-strings
• Quantum spectral curve for (q,t)-matrix model
• Geometric transition in Non-perturbative Topological string
• Note on S-channel factorization in multitrace Berenstein-Maldacena-Nastase correlators
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Free field theory and observables of periodic Macdonald processes
• Symmetric Orbifold Theories from Little String Residues
• Stable maps to Looijenga pairs: orbifold examples
• Orbifold melting crystal models and reductions of Toda hierarchy
• Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Instantons on Young diagrams with matters
• $DE$-type little strings from glued brane webs
• Topological String on Toric CY3s in Large Complex Structure Limit
• $S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants
• ADHM Wilson line defect indices
• Orthosymplectic Superinstanton Counting and Brane Dynamics
• On explicit formulae of LMOV invariants
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
• G_2 Holonomy, Mirror Symmetry and Phases of N=1 SYM
• Topological open string amplitudes on local toric del Pezzo surfaces via remodeling the B-model
• Phase space polarization and the topological string: a case study
• BPS states, crystals and matrices
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• Constant maps in equivariant topological strings and geometric modeling of fluxes
• O-vertex, O7${}^{+}$-plane, and Topological Vertex
• Open/Closed BPS Correspondence and Integrality
• Quantum mirror curve of periodic chain geometry
• Hopf superpolynomial from topological vertices
• Attractor invariants, brane tilings and crystals
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Resurgence of Refined Topological Strings and Dual Partition Functions
• Topological Field Theory Amplitudes for $A_{N-1}$ Fibration
• Towards a gauge theory interpretation of the real topological string
• A String Dual for Partially Topological Chern-Simons-Matter Theories
• Cubic Hodge integrals and integrable hierarchies of Volterra type
• Refined open topological strings revisited
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• Three-partition Hodge integrals and the topological vertex
• Elliptic Genus, Anomaly Cancellation and Heterotic M-theory
• Brane Webs and Random Processes
• Matrix model and dimensions at hypercube vertices
• A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings
• Mirror Symmetry, D-brane Superpotential and Ooguri-Vafa Invariants of Compact Calabi-Yau Manifolds
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Testing bound dark energy with cosmological parameter and fundamental constant evolution
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• The Grassmannian VOA
• Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops
• Topological String Partition Function on Generalised Conifolds
• Preons, Braid Topology, and Representations of Fundamental Particles
• BIons in topological string theory
• Generalized q-deformed Correlation Functions as Spectral Functions of Hyperbolic Geometry
• Symplectic cuts and open/closed strings I
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Compactified Webs and Domain Wall Partition Functions
• Quantum Barnes function as the partition function of the resolved conifold
• Topological Vertex, String Amplitudes and Spectral Functions of Hyperbolic Geometry
• Fermionic gluing principle of the topological vertex
• Coloured refined topological vertices and parafermion conformal field theories
• Gromov--Witten theory of $[{\mathbb{C}}^2/{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$
• Rank 2 sheaves on toric 3-folds: classical and virtual counts
• The orbifold DT/PT vertex correspondence
• Intertwining operator and integrable hierarchies from topological strings
• Free field realization of the Ding-Iohara algebra at general levels
• The universality class of the first levels in low-dimensional gravity
• Non-perturbative Symmetries of Little Strings and Affine Quiver Algebras
• Enumerative geometry of surfaces and topological strings
• The log-open correspondence for two-component Looijenga pairs
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
• On affine coordinates of the tau-function for open intersection numbers
• Quantum chaos in 2D gravity