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Virtual Euler characteristics via topological recursion

Leonid O. Chekhov

Abstract

We use Seiberg--Witten-like relations in the topological recursion framework to obtain virtual Euler characteristics for uni- and multicellular maps for ensembles of classic orthogonal polynomials and for ensembles related to nonorientable surfaces. We also discuss Harer--Zagier-type recursion relations for 1-point correlation function for the Legendre ensemble.

Virtual Euler characteristics via topological recursion

Abstract

We use Seiberg--Witten-like relations in the topological recursion framework to obtain virtual Euler characteristics for uni- and multicellular maps for ensembles of classic orthogonal polynomials and for ensembles related to nonorientable surfaces. We also discuss Harer--Zagier-type recursion relations for 1-point correlation function for the Legendre ensemble.
Paper Structure (17 sections, 3 theorems, 84 equations, 1 figure, 1 table)

This paper contains 17 sections, 3 theorems, 84 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Virtual Euler characteristic for the Gaussian model
  3. Representing Gaussian model in canonical times
  4. Relation to the virtual Euler characteristics
  5. Calculating $\varkappa_{g,s}$ for the Gaussian model
  6. $\varkappa^{\text{GUT}}_{g,s}$ via the Penner model
  7. Virtual Euler characteristics for other models
  8. The Legendre model
  9. The model for nonorientable surfaces
  10. Nonorientable surfaces via Penner-like model
  11. Nonorientable surfaces via the Selberg integral
  12. The Marchenko--Pastur ensemble
  13. One-point function for the Legendre model
  14. Harer--Zagier-like linear recursion relations
  15. Five-term recursion
  16. ...and 2 more sections

Key Result

Lemma 1

ACNP The genus-$g$ term of $s$-backbone case is given by the following (finite!) sum of homotopically equivalent diagrams (fatgraphs): where we allow all possible diagrams with vertices of order 3 and higher and we have exactly $s$ boundary components (faces). The factor $\#\hbox{Aut,} (\Gamma)$ is the standard symmetry factor and the variables $\lambda_e^{(1)}$ and $\lambda_e^{(2)}$ are $\lambda

Figures (1)

  • Figure 1: Doing a sum over ladder diagrams. New propagators are depicted as tiny ladders.

Theorems & Definitions (4)

  • Lemma 1
  • Remark 2.1
  • Lemma 2
  • Lemma 3