Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

L. Chekhov, B. Eynard

TL;DR

The paper develops a diagrammatic framework to compute the free energy of the Hermitian one-matrix model to all genera in the multi-cut regime. Building on loop equations and Bergmann-kernel technology, it extends Ey's resolvent diagram approach to arbitrary finite numbers of cuts, providing a genus-by-genus iterative construction with an inverse loop-insertion (the $H$-operator) that expresses each genus contribution as a linear action on the corresponding resolvent $W_g$. The authors give explicit planar (genus-zero) and genus-one formulas, and present a complete scheme for higher-genus terms via rooted-tree diagrams with three-valent vertices, together with a scaling relation that constrains the full free-energy functional. This work enables systematic evaluation of multicut matrix-model free energies, with applications to isomonodromic deformations, Whitham hierarchies, and Dijkgraaf–Vafa-type analyses in gauge theories.

Abstract

We present a diagrammatic technique for calculating the free energy of the Hermitian one-matrix 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).

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

TL;DR

The paper develops a diagrammatic framework to compute the free energy of the Hermitian one-matrix model to all genera in the multi-cut regime. Building on loop equations and Bergmann-kernel technology, it extends Ey's resolvent diagram approach to arbitrary finite numbers of cuts, providing a genus-by-genus iterative construction with an inverse loop-insertion (the -operator) that expresses each genus contribution as a linear action on the corresponding resolvent . The authors give explicit planar (genus-zero) and genus-one formulas, and present a complete scheme for higher-genus terms via rooted-tree diagrams with three-valent vertices, together with a scaling relation that constrains the full free-energy functional. This work enables systematic evaluation of multicut matrix-model free energies, with applications to isomonodromic deformations, Whitham hierarchies, and Dijkgraaf–Vafa-type analyses in gauge theories.

Abstract

We present a diagrammatic technique for calculating the free energy of the Hermitian one-matrix 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).

Paper Structure

This paper contains 10 sections, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Matrix models in $1/N$-expansion
  3. Loop equation and resolvents
  4. Solution in genus zero
  5. Calculating resolvents. Diagram technique
  6. Bergmann bidifferential and two-point resolvent
  7. The iterative procedure
  8. Inverting the loop insertion operator. Free energy
  9. The $H$-operator
  10. Scaling relation

Figures (1)

  • Figure 1: The arrangement of integration contours on the Riemann surface.