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Topological recursion for fully simple maps from ciliated maps

Gaëtan Borot, Séverin Charbonnier, Elba Garcia-Failde

Abstract

Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve $(y, x)$, making use of the topological recursion for ciliated maps arXiv:2105.08035.

Topological recursion for fully simple maps from ciliated maps

Abstract

Ordinary maps satisfy topological recursion for a certain spectral curve . We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve , making use of the topological recursion for ciliated maps arXiv:2105.08035.

Paper Structure

This paper contains 21 sections, 16 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction: maps and fully simple maps
  2. Multi-ciliated maps
  3. Relating fully simple maps and multi-ciliated maps
  4. Permutation model for fully simple maps
  5. Characterisation of simplicity in the permutation model
  6. Characterisation of full simplicity in the permutation model
  7. Characterisation of multi-ciliated maps in the permutation model
  8. Star constraint
  9. Uniqueness constraint
  10. Topological recursion for fully simple maps
  11. Topological recursion for ciliated maps
  12. Relating fully simple and ciliated generating series
  13. Identification of the spectral curve
  14. Applications
  15. Symplectic invariance
  16. ...and 6 more sections

Key Result

Lemma 2.8

BCEG-F If $k_1 \geq 2$, set $\underline{k}'=(k_1-1,k_2,\ldots,k_n)$. We have: $\star$

Figures (1)

  • Figure 1: The left panel depicts the local structure of an edge in a map. The oriented edges $e_1$ and $e_2$ are indicated by the arrows. With our conventions, $e_i$ is adjacent to face $f_i$ and incident to vertex $v_i$ for $i = 1, 2$. The right panel depicts the local structure of a vertex in a map, including the action of the permutations $\sigma_0, \sigma_1, \sigma_2$ on an oriented edge $e$.

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: Generating series of ordinary and fully simple maps
  • Definition 2.1: Constraints on the vertices
  • Definition 2.2: Ciliated maps
  • Definition 2.3: Multi-ciliated maps
  • Definition 2.4: Degree of a map
  • Definition 2.5: Local weights
  • Definition 2.6
  • Definition 2.7: Generating series of (multi-)ciliated maps
  • ...and 29 more