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

We compute the orbifold Euler characteristics of ${\overline{\mathcal{M}}}_{g,n}$ by applying the formalisms developed in (Wang et al., J. High Energy Phys. 2019(4):135, 2019; Zhou, arXiv:1412.1604, 2014).We take the works of Harer–Zagier (Invent. Math. 85(3):457–485, 1986) and Bini–Harer (J. Eur. Math. Soc. 13(2):487–512, 2011) as the starting point, and prove two types of recursion relations to compute $\chi ({\overline{\mathcal{M}}}_{g,n})$. As applications of these recursions, we give some numerical data and derive some closed formulas, and generalize Manin’s functional equation for $\chi ({\overline{\mathcal{M}}}_{0,n})$ to higher genera cases. Moreover, in genus zero the results are related to Ramanujan polynomials. We also show that the generating series of $\chi ({\overline{\mathcal{M}}}_{g,n})$ is the logarithm of a particular tau-function of KP hierarchy evaluated at times specified by the generating series of $\chi ({\mathcal{M}}_{g,n})$.