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The Laplace transform of the cutandjoin equation and the BouchardMarino conjecture on Hurwitz numbers
"... Abstract. We calculate the Laplace transform of the cutandjoin equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the WeilPetersson volume of the moduli space of bordered hyperbolic surfa ..."
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Cited by 44 (16 self)
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Abstract. We calculate the Laplace transform of the cutandjoin equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the WeilPetersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert Wfunction is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory. Contents
Algebraic methods in random matrices and enumerative geometry
, 2008
"... We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defines a sequence of differential forms, and a sequence of complex numbers Fg. We recall the definitio ..."
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Cited by 38 (9 self)
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We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defines a sequence of differential forms, and a sequence of complex numbers Fg. We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several examples of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, nonintersecting brownian motions,...
A holomorphic and background independent partition function for matrix models and topological strings
, 2009
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The D5brane effective action and superpotential in N = 1 Compactifications
, 2008
"... The fourdimensional effective action for D5branes in generic compact CalabiYau orientifolds is computed by performing a KaluzaKlein reduction. The N = 1 Kähler potential, the superpotential, the gaugekinetic coupling function and the Dterms are derived in terms of the geometric data of the int ..."
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Cited by 37 (4 self)
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The fourdimensional effective action for D5branes in generic compact CalabiYau orientifolds is computed by performing a KaluzaKlein reduction. The N = 1 Kähler potential, the superpotential, the gaugekinetic coupling function and the Dterms are derived in terms of the geometric data of the internal space and of the twocycle wrapped by the D5brane. In particular, we obtain the D5brane and flux superpotential by integrating out fourdimensional threeforms which couple via the ChernSimons action. Also the infinitesimal complex structure deformations of the twocycle induced by the deformations of the ambient space contribute to the Fterms. The superpotential can be expressed in terms of relative periods depending on both the open and closed moduli. To analyze this dependence we blow up along the twocycle and obtain a rigid divisor in an auxiliary compact threefold with negative first Chern class. The variation of the mixed Hodge structure on this blownup geometry is equivalent to the original deformation problem and can be analyzed by PicardFuchs equations. We exemplify the blowup procedure for a noncompact CalabiYau threefold given by the canonical bundle over del Pezzo surfaces.
Nonperturbative Effects and the Large–Order Behavior of Matrix Models and Topological Strings
, 711
"... Abstract: This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large–order behavior of perturbation theory. We study instanton configurations in generic one–cut matrix models, obtaining explicit results for the one–instanton amplitude ..."
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Cited by 33 (6 self)
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Abstract: This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large–order behavior of perturbation theory. We study instanton configurations in generic one–cut matrix models, obtaining explicit results for the one–instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi–Yau manifolds. This yields very precise predictions for the large–order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two–dimensional gravity, we also obtain in this way the large–order asymptotics of the relevant solution to the Painlevé I equation, including corrections in inverse genus. From a mathematical point of view, our results predict
Unquenched flavor and tropical geometry in strongly coupled Chern–Simons–matter theories
, 2011
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Holomorphic Anomaly and Matrix Models
, 2007
"... The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vaf ..."
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Cited by 29 (10 self)
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The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic 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.