Volumes of odd strata of quadratic differentials

TL;DR

This paper provides a new formula expressing the volumes of odd strata of quadratic differentials as sums over stable graphs, linking intersection theory, ribbon graphs, and volume calculations.

Contribution

It generalizes existing volume formulas for principal strata to odd strata, involving intersection numbers and combinatorial classes.

Findings

Derived a sum-over-graphs formula for volumes of odd quadratic differential strata.

Connected volume computations with intersection theory and ribbon graph enumeration.

Proposed conjectures on large genus asymptotics based on the new formula.

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.