Holomorphic anomaly and matrix models

Abstract

The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf--Vafa conjecture relating matrix models to type B topological strings on certain local Calabi--Yau threefolds.

23 pages, LaTex, 3 figures

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• Integrability of the holomorphic anomaly equations
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• Algebraic methods in random matrices and enumerative geometry
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
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• Extended Holomorphic Anomaly and Loop Amplitudes in Open Topological String
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• The uses of the refined matrix model recursion
• Intersection numbers of spectral curves
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Virasoro constraints for Kontsevich-Hurwitz partition function
• ABJM Wilson loops in the Fermi gas approach
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• On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
• Holomorphic Anomaly and Quantum Mechanics
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• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• Direct Integration and Non-Perturbative Effects in Matrix Models
• Evidence for Tadpole Cancellation in the Topological String
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• A short overview of the "Topological recursion"
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• Geometrical interpretation of the topological recursion, and integrable string theories
• On gauge/string correspondence and mirror symmetry
• Challenges of beta-deformation
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• Topological expansion of the chain of matrices
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• D-instanton, threshold corrections, and topological string
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• The topological open string wavefunction
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• The holomorphic anomaly for open string moduli
• A Lagrangean formalism for Hermitean matrix models
• Stable quotients and the holomorphic anomaly equation
• Orbifold Euler Characteristics of ${\overline{\mathcal{M}}}_{g,n}$
• One-cut solution of the $\beta$-ensembles in the Zhukovsky variable
• Large N instantons from topological strings
• Topological open string amplitudes on local toric del Pezzo surfaces via remodeling the B-model
• Dijkgraaf-Vafa conjecture and beta-deformed matrix models
• The Laplace transform of the cut-and-join equation of Mariño-Vafa formula and its applications
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• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Aspects géométriques et intégrables des modèles de matrices aléatoires
• Fourier-Like Transforms of Stable Graphs and Holomorphic Anomaly Equations
• Geometric Transition as a Change of Polarization