Topological expansion and boundary conditions

@article{Eynard2007TopologicalEA,
  title={Topological expansion and boundary conditions},
  author={Bertrand Eynard and Nicolas Orantin},
  journal={Journal of High Energy Physics},
  year={2007},
  volume={2008},
  pages={037-037}
}
In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words 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. 

Figures from this paper

CFT and topological recursion

We study the quasiclassical expansion associated with a complex curve. In a more specific context this is the 1/N expansion in U(N)-invariant matrix integrals. We compare two approaches, the CFT

Enumeration of maps with self-avoiding loops and the model on random lattices of all topologies

We compute the generating functions of an model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, the topologies were already known, and here we compute all the

Topological recursion in enumerative geometry and random matrices

We review the method of symplectic invariants recently introduced to solve matrix models' loop equations in the so-called topological expansion, and further extended beyond the context of matrix

Algebraic methods in random matrices and enumerative geometry

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

Non-homogenous disks in the chain of matrices

A bstractWe investigate the generating functions of multi-colored discrete disks with non-homogenous boundary condition in the context of the Hermitian multi-matrix model where the matrices are

Liouville theory and random maps

This thesis explore several aspects of random maps through the study of three models. First, we examine the properties of a measure defined on the set of planar Delaunay triangulations with n

Complete solution of the LSZ Model via Topological Recursion

. We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological

Matrix Field Theory

This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over

Minimal open strings

We study FZZT-branes and open string amplitudes in (p, q) minimal string theory. We focus on the simplest boundary changing operators in two-matrix models, and identify the corresponding operators in

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) β-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and

References

SHOWING 1-10 OF 24 REFERENCES

Free energy topological expansion for the 2-matrix model

We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/N expansion of the

Topological expansion of mixed correlations in the Hermitian 2-matrix model and x–y symmetry of the Fg algebraic invariants

We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e. a generating function for counting bicolored discrete surfaces with non-uniform boundary conditions.

Topological expansion for the 1-hermitian matrix model correlation functions

We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the

All genus correlation functions for the hermitian 1-matrix model

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N2 expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic

Invariants of algebraic curves and topological expansion

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.

Combinatorial solution of the two-matrix model

Large N expansion of the 2-matrix model, multicut case

We present a method, based on loop equations, to compute recursively, all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model, in the case where the

2D gravity and random matrices