Hermitian matrix model free energy: Feynman graph technique for all genera

Chekhov, Leonid; Eynard, Bertrand

doi:10.1088/1126-6708/2006/03/014

Cited by 187 publications

(365 citation statements)

References 38 publications

Supporting

Mentioning

337

Contrasting

Unclassified

Order By: Relevance

“…In [43], the methods of [40,41,64] are used to evaluate the genus one and two terms of the matrix model free energy for the class of double-scaling limits considered in [4]. The simplest case corresponds to the above conifold singularity with two branch points colliding and no double points, n = 2, m = 0, and the results match the genus zero, one and two terms in (4.12).…”

Section: Simple Example: Two Branch Pointsmentioning

confidence: 91%

“…The most powerful methods for calculating the F g involves using orthogonal polynomials which we review in Section 4.2; however, as we shall see, these techniques appear only to be successful for the cases with one or two branch points and any additional number of double points. The only known way to calculate the F g in general involves analyzing the loop equations and in particular using the algorithms recently developed in [40,41]. We use these techniques to prove our scaling ansatz (2.12) in Section 4.3.…”

Section: The Double-scaling Limit Of the Free Energymentioning

confidence: 99%

“…In order to find an expression for F g , g > 1, one should consider the same graphs relevant for W (g) (x 1 ) and do then the following [41]:…”

Section: The Loop Equationsmentioning

confidence: 99%

“…Note that when evaluating the final residues at x 2 = σ i , one needs to expand the above integral by setting q 0 = σ i [41]. It is also understood that the evaluation of the residues starts from the outer branches and proceeds towards the root.…”

Section: The Loop Equationsmentioning

confidence: 99%

See 3 more Smart Citations

Double scaling limits in gauge theories and matrix models

Bertoldi¹,

Hollowood²,

Miramontes³

2006

J. High Energy Phys.

View full text Add to dashboard Cite

Abstract:We show that N = 1 gauge theories with an adjoint chiral multiplet admit a wide class of large-N double-scaling limits where N is taken to infinity in a way coordinated with a tuning of the bare superpotential. The tuning is such that the theory is near an Argyres-Douglas-type singularity where a set of non-local dibaryons becomes massless in conjunction with a set of confining strings becoming tensionless. The doubly-scaled theory consists of two decoupled sectors, one whose spectrum and interactions follow the usual large-N scaling whilst the other has light states of fixed mass in the large-N limit which subvert the usual large-N scaling and lead to an interacting theory in the limit. F -term properties of this interacting sector can be calculated using a Dijkgraaf-Vafa matrix model and in this context the doublescaling limit is precisely the kind investigated in the "old matrix model" to describe two-dimensional gravity coupled to c < 1 conformal field theories. In particular, the old matrix model double-scaling limit describes a sector of a gauge theory with a mass gap and light meson-like composite states, the approximate Goldstone boson of superconformal invariance, with a mass which is fixed in the double-scaling limit. Consequently, the gravitational F -terms in these cases satisfy the string equation of the KdV hierarchy.

show abstract

Section: Simple Example: Two Branch Pointsmentioning

confidence: 91%

Section: The Double-scaling Limit Of the Free Energymentioning

confidence: 99%

“…In order to find an expression for F g , g > 1, one should consider the same graphs relevant for W (g) (x 1 ) and do then the following [41]:…”

Section: The Loop Equationsmentioning

confidence: 99%

Section: The Loop Equationsmentioning

confidence: 99%

See 2 more Smart Citations

Double scaling limits in gauge theories and matrix models

Bertoldi¹,

Hollowood²,

Miramontes³

2006

J. High Energy Phys.

View full text Add to dashboard Cite

show abstract

“…We leave to a future investigation the simultaneous integration of (5.17) to get a closed formula for F g in terms of W g 1 n 1 's. For usual maps, this step was performed systematically in [Che06], but the problem here seems more complicated since the evaluation of ω g n pz 1 , . .…”

Section: Generating Series Of Closed Stuffed Mapsmentioning

confidence: 99%

Formal multidimensional integrals, stuffed maps, and topological recursion

Borot

2014

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

View full text Add to dashboard Cite

We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [EO07a] with initial condition. In terms of a 1d gas of eigenvalues, this model includes -on top of the squared Vandermonde -multilinear interactions of any order between the eigenvalues. In this problem, the initial data pω In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps". In a sense, our results complete the program of the "moment method" initiated in the 90s to compute the formal 1{N in the one hermitian matrix model.

show abstract

Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

Mariño

2014

Fortschritte der Physik

237

293

View full text Add to dashboard Cite

In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons, and then present a detailed review of non-perturbative effects in matrix models. Finally, I consider two applications of these techniques in string theory

show abstract

Hermitian matrix model free energy: Feynman graph technique for all genera

Abstract: SPhT-T05/046 ITEP/TH-??/05We present the diagrammatic technique for calculating the free energy of the Hermitian onematrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

Cited by 187 publications

References 38 publications

Double scaling limits in gauge theories and matrix models

Double scaling limits in gauge theories and matrix models

Formal multidimensional integrals, stuffed maps, and topological recursion

Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

Contact Info

Product

Resources

About