Orbifold Euler Characteristics of $\overline{\mathcal M}_{g,n}$

Abstract

We solve the problem of the computation of the orbifold Euler characteristics of $\Mbar_{g,n}$. We take the works of Harer-Zagier \cite{hz} and Bini-Harer \cite{bh} as our starting point, and apply the formalisms developed in \cite{wz} and \cite{zhou1} to this problem. These formalisms are typical examples of mathematical methods inspired by quantum field theories. We also present many closed formulas and some numerical data. In genus zero the results are related to Ramanujan polynomials, and in higher genera we get recursion relations almost identical to the recursion relations for Ramanujan polynomials but with different initial values. We also show that the generating series given by the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$ is the logarithm of the KP tau-function of the topological 1D gravity evaluated at the times given by the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$. Conversely, the logarithm of this tau-function evaluated at the times given by certain generating series of the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$ is a generating series of the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$. This is a new example of open-closed duality.