Papers by Bertrand Eynard
Journal of High Energy Physics, Jun 13, 2008
In this article, we compute the topological expansion of all possible mixed-traces in a hermitian... more In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words we give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.
Journal of High Energy Physics, Dec 21, 2005
We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N ... more We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N 2 expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.
Journal of High Energy Physics, Dec 19, 2006
We compute the complete topological expansion of the formal hermitian twomatrix model. For this, ... more We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.
arXiv (Cornell University), May 5, 2012
, proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calab... more , proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been verified to low genus for several toric CY3folds, and proved to all genus only for C 3 . In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model.Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in 2 steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to special geometry property implied by the topological recursion.
arXiv (Cornell University), Nov 13, 2013
The goal of this article is to rederive the connection between the Painlevé 5 integrable system a... more The goal of this article is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ -function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N -5 ).
Communications in Number Theory and Physics, 2015
We formulate a notion of "abstract loop equations," and show that their solution is provided by a... more We formulate a notion of "abstract loop equations," and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one-and two-Hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N ) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
Journal of Geometry and Physics, Feb 1, 2011
We introduce a new matrix model representation for the generating function of simple Hurwitz numb... more We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in . As an application, we prove the conjecture proposed by Bouchard and Mariño [2], relating Hurwitz numbers to the spectral invariants of the Lambert curve e x = ye -y .
HAL (Le Centre pour la Communication Scientifique Directe), May 7, 2007
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their p... more For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition, we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a τ -function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e., the (p, q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV τ -function.
arXiv (Cornell University), Mar 23, 2013
We formulate a notion of "abstract loop equations", and show that their solution is provided by a... more We formulate a notion of "abstract loop equations", and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpN q Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
arXiv (Cornell University), Mar 8, 2010
We construct a matrix model that reproduces the topological string partition function on arbitrar... more We construct a matrix model that reproduces the topological string partition function on arbitrary toric Calabi-Yau 3-folds. This demonstrates, in accord with the BKMP "remodeling the B-model" conjecture, that Gromov-Witten invariants of any toric Calabi-Yau 3-fold can be computed in terms of the spectral invariants of a spectral curve. Moreover, it proves that the generating function of Gromov-Witten invariants is a tau function for an integrable hierarchy. In a follow-up paper, we will explicitly construct the spectral curve of our matrix model and argue that it equals the mirror curve of the toric Calabi-Yau manifold.
HAL (Le Centre pour la Communication Scientifique Directe), May 7, 2007
We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e... more We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e. generating function for counting bicolored discrete surfaces with non uniform boundary conditions. As an application, we prove the x -y symmetry of .
arXiv (Cornell University), Jun 8, 2021
We prove that the topological recursion formalism can be used to quantize any generic classical s... more We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.
In this article, we compute the topological expansion of all possible mixed-traces in a hermitian... more In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words we give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.
Random Matrices: Theory and Applications, 2014
The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and... more The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N-5).
Annales Henri Poincaré, 2017
Starting from a d × d rational Lax pair system of the form ∂ x Ψ = LΨ and ∂ t Ψ = RΨ we prove tha... more Starting from a d × d rational Lax pair system of the form ∂ x Ψ = LΨ and ∂ t Ψ = RΨ we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the "topological type property". A consequence is that the formal -WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems.
Journal of High Energy Physics, 2012
We write the loop equations for the β two-matrix model, and we propose a topological recursion al... more We write the loop equations for the β two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a “quantum” spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study the case where that quantum spectral curve is completely degenerate, it satisfies a Bethe ansatz, and the spectral curve is the Baxter TQ relation.
Journal of High Energy Physics, 2006
We compute the complete topological expansion of the formal hermitian twomatrix model. For this, ... more We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.
Journal of High Energy Physics, 2005
We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N ... more We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N 2 expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.
Communications in Number Theory and Physics, 2015
We formulate a notion of "abstract loop equations", and show that their solution is provided by a... more We formulate a notion of "abstract loop equations", and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpN q Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
Journal of High Energy Physics, 2008
In this article, we compute the topological expansion of all possible mixed-traces in a hermitian... more In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words we give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.
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Papers by Bertrand Eynard