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Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces

S. Barazer

TL;DR

This work develops a comprehensive framework for volumes of moduli spaces of directed metric ribbon graphs by introducing acyclic decompositions via admissible multi-curves. It establishes a canonical, vertex-wise surgery process that yields a directed stable graph encoding the surgeries, and proves that the resulting decomposition is acyclic and unique under a fixed vertex order. This leads to a Mirzakhani-type recursion for volumes V_{g,n^+,n^-}(L^+|L^-) and, in particular, a closed-form recursion in the four-valent case, as well as a Cut-and-Join-type equation in the one-negative-boundary scenario linked to Grothendieck dessins d’enfants. The approach hinges on a Kontsevich-style two-form and a robust interplay between combinatorial Teichmüller theory, measured foliations, and the cohomology of ribbon graphs, producing piecewise polynomial volume formulas with explicit wall structures. Overall, the paper provides a principled, scalable method to compute volumes of directed moduli spaces and connects these volumes to broader enumerative-geometric phenomena such as dessins d’enfants and topological recursion.

Abstract

In this paper we study metric ribbon graphs, in particular, directed metric ribbon graphs. These ribbon graphs are dual to bipartite maps and appear in the context of Abelian differentials. We prove that it is possible to decompose a directed ribbon graph into a family of ribbon graphs with one vertex, by performing surgeries along appropriate multi curves. The decomposition is canonical and we call it acyclic decomposition due to a condition on the stable graphs that encode the surgeries. This result provides a recursion scheme for volumes of moduli space of directed metric ribbon graphs, we give explicitly the recursion in the case of four valent metric ribbon graphs. In a particular case, we give applications to count of Grothendieck dessins d'enfants.

Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces

TL;DR

This work develops a comprehensive framework for volumes of moduli spaces of directed metric ribbon graphs by introducing acyclic decompositions via admissible multi-curves. It establishes a canonical, vertex-wise surgery process that yields a directed stable graph encoding the surgeries, and proves that the resulting decomposition is acyclic and unique under a fixed vertex order. This leads to a Mirzakhani-type recursion for volumes V_{g,n^+,n^-}(L^+|L^-) and, in particular, a closed-form recursion in the four-valent case, as well as a Cut-and-Join-type equation in the one-negative-boundary scenario linked to Grothendieck dessins d’enfants. The approach hinges on a Kontsevich-style two-form and a robust interplay between combinatorial Teichmüller theory, measured foliations, and the cohomology of ribbon graphs, producing piecewise polynomial volume formulas with explicit wall structures. Overall, the paper provides a principled, scalable method to compute volumes of directed moduli spaces and connects these volumes to broader enumerative-geometric phenomena such as dessins d’enfants and topological recursion.

Abstract

In this paper we study metric ribbon graphs, in particular, directed metric ribbon graphs. These ribbon graphs are dual to bipartite maps and appear in the context of Abelian differentials. We prove that it is possible to decompose a directed ribbon graph into a family of ribbon graphs with one vertex, by performing surgeries along appropriate multi curves. The decomposition is canonical and we call it acyclic decomposition due to a condition on the stable graphs that encode the surgeries. This result provides a recursion scheme for volumes of moduli space of directed metric ribbon graphs, we give explicitly the recursion in the case of four valent metric ribbon graphs. In a particular case, we give applications to count of Grothendieck dessins d'enfants.
Paper Structure (112 sections, 99 theorems, 299 equations, 29 figures)

This paper contains 112 sections, 99 theorems, 299 equations, 29 figures.

Table of Contents

  1. Introduction:
  2. Directed ribbon graphs, moduli spaces, and their volumes:
  3. Surgeries on directed ribbon graphs and acyclic decompositions:
  4. Recursion for the volumes:
  5. Special case, Cut-and-Join equation and Grothendieck dessins d’enfants:
  6. Acknowledgment:
  7. Plan of the paper:
  8. Topological background and notations:
  9. Surfaces and curves
  10. Surfaces, directed surfaces and decorations:
  11. Curves and multi-curves:
  12. Directed multi-curves:
  13. Decoration:
  14. Multi-arcs:
  15. Foliations, quadratic differentials and Abelian differentials:
  16. ...and 97 more sections

Key Result

Theorem 1

The functions $V_{g,n^+,n^-}(L^+|L^-)$ are continuous, homogeneous, piecewise polynomials of degree $4g-3+n^++n^-$ defined on $\Lambda_{n^+,n^-}$.

Figures (29)

  • Figure 1: A directed ribbon graph on a directed surface of type $(0,2,1)$
  • Figure 2: A directed acyclic stable graph.
  • Figure 3: An acyclic decomposition.
  • Figure 4: The four terms of the recursion in theorem \ref{['intro_thm_acycl_curve']}
  • Figure 5: Directed pants of type $(0,2,1)$ and $(0,1,2)$.
  • ...and 24 more figures

Theorems & Definitions (200)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 1
  • Definition 1
  • Remark 2: Orientation of the boundary components
  • Remark 3: marked points and labelling:
  • ...and 190 more