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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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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,...
All orders asymptotic expansion of large partitions
, 2008
"... The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and ..."
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Cited by 28 (6 self)
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The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the GromovWitten invariants of the Xp = O(p − 2) ⊕ O(−p) → P1 CalabiYau 3fold, and we prove a conjecture of M. Mariño, that the generating functions Fg of Gromov–Witten invariants of Xp, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.
Topological expansion of the Bethe ansatz, and noncommutative algebraic geometry
 JHEP 0903, 094 (2009) [arXiv:0809.3367 [mathph
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 21 (5 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. SPhTT08/140 Topological expansion of the Bethe ansatz, and noncommutative algebraic geometry
The volume conjecture, perturbative knot invariants, and recursion relations for topological strings
 Nuclear Phys. B 849
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Quantum Curves and DModules
, 2008
"... In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an Ibrane configuration, which consists of D4 and D6branes intersecting along a holomorphic curve in a complex surface, together with a Bfield. Mathematically, it is described ..."
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In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an Ibrane configuration, which consists of D4 and D6branes intersecting along a holomorphic curve in a complex surface, together with a Bfield. Mathematically, it is described by a holonomic Dmodule. Here we focus on spectral curves, which play a prominant role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the Ibrane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, our formalism elegantly reconstructs the complete dual NekrasovOkounkov partition function from a quantum SeibergWitten curve.
The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures
"... Abstract. We derive the spectral curves for qpart double Hurwitz numbers, rspin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition fun ..."
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Abstract. We derive the spectral curves for qpart double Hurwitz numbers, rspin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.
Toda Theories, Matrix Models, Topological Strings, and N = 2 Gauge Systems
, 909
"... We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an An−1 singularity over a Riemann surface. These models realize the N = 2 SU(n) superconformal gauge systems recently studied by Gaiotto and collaborators. Employing large N duali ..."
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We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an An−1 singularity over a Riemann surface. These models realize the N = 2 SU(n) superconformal gauge systems recently studied by Gaiotto and collaborators. Employing large N dualities we show why the partition function of topological strings in these backgrounds is captured by the chiral blocks of An−1 Toda systems and derive the dictionary recently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zero Riemann surfaces, we show how these systems can also be realized by Pennerlike matrix models with logarithmic potentials. The SeibergWitten curve can be understood as the spectral curve of these matrix models which arises holographically at large N. In this context the Nekrasov deformation maps to the βensemble of generalized matrix models, that in turn maps to the Toda system with general background charge. We also point out the notion of a double holography for this system, when both n and N are large. September