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

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.