[Paper Review] Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials

D. Chen, Martin Moeller+4 more|arXiv (Cornell University)|Dec 4, 2019

Mathematical Dynamics and Fractals31 references9 citations

TL;DR

This paper proposes and proves a conjectural formula for Masur-Veech volumes of strata of quadratic differentials using intersection theory on moduli spaces of curves. It establishes that for principal strata with only simple zeros, the volume equals an explicit intersection number involving the top Segre class of the quadratic Hodge bundle, which reduces to linear Hodge integrals. The key result is a proof that this class can be computed via Eynard-Orantin topological recursion, enabling explicit computation of volumes and related invariants.

ABSTRACT

We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compute the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with $\psi$-classes can be computed by Eynard-Orantin topological recursion. As applications, we analyze numerical properties of Masur-Veech volumes, area Siegel-Veech constants and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux-Hubert, Fougeron, and elaborated in [the7]. We also describe conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

Motivation & Objective

• To conjecture and prove a formula for Masur-Veech volumes of strata of quadratic differentials using intersection numbers on moduli spaces.
• To establish that for principal strata (with only simple zeros), the volume reduces to computing the top Segre class of the quadratic Hodge bundle.
• To show that this Segre class intersection with ψ-classes can be computed via Eynard-Orantin topological recursion.
• To apply the formula to analyze asymptotic properties of volumes, Siegel-Veech constants, and Lyapunov exponents for fixed genus and varying number of zeros.
• To verify conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents in arbitrary affine invariant submanifolds, focusing on the principal strata case.

Proposed method

• Proposes a conjectural formula (Conjecture 1.1) expressing Masur-Veech volumes as intersection numbers involving the first Chern class ζ of the tautological line bundle and ψ-classes on the projectivized stratum.
• Proves the formula for strata with only odd-order zeros (Theorem 1.2), showing the volume equals an intersection number involving only ζ and no ψ-classes.
• For principal strata with only simple zeros, identifies the top Segre class of the quadratic Hodge bundle Qg,n as the relevant intersection class.
• Reduces the volume computation to linear Hodge integrals via known formulas for characteristic classes of the Hodge bundle (Theorem 1.3).
• Uses the Eynard-Orantin topological recursion on a specific spectral curve (x(z) = -z - ln z, y(z) = z²) to compute the intersection of the Segre class with ψ-classes.
• Constructs a class Ωg,n[a,b] via asymptotic expansion of the recursion kernel, showing it generates the relevant intersection numbers.

Experimental results

Research questions

• RQ1 Can Masur-Veech volumes of strata of quadratic differentials be computed via intersection numbers on the moduli space of curves?
• RQ2 Is the top Segre class of the quadratic Hodge bundle computable via topological recursion, and does it yield the Masur-Veech volume for principal strata?
• RQ3 What are the asymptotic behaviors of Masur-Veech volumes, Siegel-Veech constants, and sums of Lyapunov exponents for principal strata as the number of zeros increases?
• RQ4 Can the intersection of the Segre class with ψ-classes be computed using Eynard-Orantin topological recursion?
• RQ5 Do the conjectural formulas for Siegel-Veech constants and Lyapunov exponents in arbitrary affine invariant submanifolds hold true for the principal strata?

Key findings

• The conjectural formula for Masur-Veech volumes (Conjecture 1.1) is proven for strata with only odd-order zeros (Theorem 1.2).
• For principal strata Qg,4g−4+2n(14g−4+n, −1n), the volume is given by an explicit intersection number involving linear Hodge integrals (Theorem 1.3).
• The intersection of the Segre class s(Qg,n) with ψ-classes is computed via Eynard-Orantin topological recursion on the spectral curve (x(z) = -z - ln z, y(z) = z²), proving the formula (4) in the appendix.
• The topological recursion yields the Masur-Veech volume as the ki=0 term of the amplitudes Fg,n[0,...,0], matching known results from [the7].
• The formula for the volume is used to prove that the normalized volume is a rational function in n of degree at most ⌊(g-1)/2⌋, confirming a conjecture from [the7].
• The paper verifies the conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents in the principal strata case, providing explicit expressions in terms of Hodge integrals.