A geometric parametrization for the virtual Euler characteristic for the moduli spaces of real and complex algebriac curves

TL;DR

This paper introduces a geometric parametrization of the virtual Euler characteristic for moduli spaces of algebraic curves, unifying real and complex cases via a polynomial in a parameter related to matrix models and symmetric functions.

Contribution

It proposes a polynomial framework connecting real and complex moduli space invariants through a geometric parameter, suggesting a new interpretation of the virtual Euler characteristic.

Findings

The virtual Euler characteristic can be expressed as a polynomial in 1/γ.

Specializations at γ=1 and γ=1/2 recover complex and real cases.

The polynomial may represent the Euler characteristic of an as-yet-unknown moduli space.

Abstract

We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/\gamma$ where $\gamma$ permits specialization, through $\gamma=1,$ to the complex case treated by Harer and Zagier and, through $\gamma=1/2$, to the real case. This polynomial appears to have geometric significance, and may be the virtual Euler characteristic of some moduli space, as yet unidentified. This is related to a conjecture that the indeterminate $b=\gamma^1-1$ is associated with a combinatorial invariant of cell-decompositions through matrix models and the Jack symmetric functions. The development uses Strebel differentials to triangulate the moduli spaces, and the identification of $\gamma$ both as a parameter in a Jack symmetric function and as a parameter in a matrix model through generalized Selberg integrals.