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

Goulden, I. P.; Harer, J. L.; Jackson, D. M.

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

Authors

Goulden, I. P.

Harer, J. L.

Jackson, D. M.

Description

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 $1/γ$ where $γ$ permits specialization, through $γ=1,$ to the complex case treated by Harer and Zagier and, through $γ=1/2$, 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 $b=γ^1-1$ 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 $γ$ both as a parameter in a Jack symmetric function and as a parameter in a matrix model through generalized Selberg integrals.
24 pages