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Volumes of odd strata of quadratic differentials

Eduard Duryev, Elise Goujard, Ivan Yakovlev

TL;DR

This work advances the quantitative study of odd strata of quadratic differentials by expressing their Masur--Veech volumes as completed sums over decorated stable graphs, with coefficients given by psi-class intersections on Witten--Kontsevich combinatorial cycles. It extends the DGZZ framework from principal strata to strata with only odd singularities by counting square-tiled surfaces via metric ribbon graphs, and carefully analyzes wall phenomena to account for degenerations that generate adjacent-strata contributions. Central to the approach are Kontsevich polynomials, the Z-operator encoding zeta-values, and a detailed treatment of metric polyhedra on non-bipartite ribbon graphs, including wall-wise polynomiality and degeneration contributions. The paper also provides a concrete example, discusses local-to-global assembly, and outlines conjectures for large-genus volume asymptotics, highlighting potential pathways to recursive relationships among volumes across strata and a deeper understanding of cylinder distributions in random square-tiled surfaces.

Abstract

We express the Masur--Veech volumes of "completed" strata of quadratic differentials with only odd singularities as a sum over stable graphs. This formula generalizes the formula of Delecroix-Goujard-Zograf-Zorich for principal strata. The coefficients of the formula are in our case intersection numbers of psi classes with the Witten-Kontsevich combinatorial classes; they naturally appear in the count of metric ribbon graphs with prescribed odd valencies. The "completed" strata that we consider are unions of odd strata and some adjacent strata, that contribute to the Masur--Veech volume with explicit weights. We present several conjectures on the large genus asymptotics of these Masur--Veech volumes that could be tackled with this formula.

Volumes of odd strata of quadratic differentials

TL;DR

This work advances the quantitative study of odd strata of quadratic differentials by expressing their Masur--Veech volumes as completed sums over decorated stable graphs, with coefficients given by psi-class intersections on Witten--Kontsevich combinatorial cycles. It extends the DGZZ framework from principal strata to strata with only odd singularities by counting square-tiled surfaces via metric ribbon graphs, and carefully analyzes wall phenomena to account for degenerations that generate adjacent-strata contributions. Central to the approach are Kontsevich polynomials, the Z-operator encoding zeta-values, and a detailed treatment of metric polyhedra on non-bipartite ribbon graphs, including wall-wise polynomiality and degeneration contributions. The paper also provides a concrete example, discusses local-to-global assembly, and outlines conjectures for large-genus volume asymptotics, highlighting potential pathways to recursive relationships among volumes across strata and a deeper understanding of cylinder distributions in random square-tiled surfaces.

Abstract

We express the Masur--Veech volumes of "completed" strata of quadratic differentials with only odd singularities as a sum over stable graphs. This formula generalizes the formula of Delecroix-Goujard-Zograf-Zorich for principal strata. The coefficients of the formula are in our case intersection numbers of psi classes with the Witten-Kontsevich combinatorial classes; they naturally appear in the count of metric ribbon graphs with prescribed odd valencies. The "completed" strata that we consider are unions of odd strata and some adjacent strata, that contribute to the Masur--Veech volume with explicit weights. We present several conjectures on the large genus asymptotics of these Masur--Veech volumes that could be tackled with this formula.

Paper Structure

This paper contains 36 sections, 37 theorems, 148 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Context
  3. Illustrative example: stratum $\mathcal{Q}(3, -1^3)$
  4. Outline of the paper
  5. Aknowledgments
  6. Volume formula
  7. Combinatorial moduli space and Kontsevich polynomial
  8. Decorated stable graphs
  9. Formula
  10. Metric ribbon graphs and counting functions
  11. Counting functions for metric ribbon graphs
  12. Passing to the dual ribbon graphs
  13. Weight functions on non-bipartite ribbon graphs
  14. Polytopes of metrics
  15. Walls and piecewise (quasi-)polynomiality
  16. ...and 21 more sections

Key Result

Theorem 2.6

For any odd composition ${\underline{k}}=(k_1, \dots, k_r)$ of $4g-4$ with $r\geq 3$ and ${k}_i\geq -1$, where the coefficients $C_{\underline g, \underline g_i}$ are obtained as follows. Let $[n], \overline{[n]}, n \ge -1$, be non-commuting variables. Declare and The coefficients of the formula are obtained by applying the following change of variables in $\overline\operatorname{Vol}(\mathcal{

Figures (7)

  • Figure 1: Ribbon graph and count of integer metrics.
  • Figure 2: Values of the counting function for integer metrics on the parameter space.
  • Figure 3: List of all decorated stable graphs in $\mathcal{G}_{1,3}^{[5,1^3]}$. The genus $g_v$ is indicated above each vertex $v$.
  • Figure 4: A degeneration of a ribbon graph $G$ in $\mathcal{RG}^{{\underline{\kappa}},*}_{g,n}$ into $9$ components. The zero-weight static bridges are dotted. The unique non-bipartite component $G_0$ has 4 faces of degrees $7,5,5,3$. Two "branches" are glued to the corners of one of the faces of degree $5$. Their boundaries (in grey) form together a face of the initial graph $G$.
  • Figure 5: An admissible joining (top) and the corresponding sequence of markers (bottom) produced by Algorithm \ref{['alg:branchToSequence']}.
  • ...and 2 more figures

Theorems & Definitions (86)

  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3: Weight of a static edge
  • proof
  • ...and 76 more