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/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.