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

Abstract

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 $\chi ({{\mathcal {M}}}_{g,n})$.

Keywords

Moduli spaces of stable curves / Orbifold Euler characteristics / Recursions / KP hierarchy / 14H10 / 14H81 / 37K25

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Funding

National Natural Science Foundation of China(12371254)

RIGHTS & PERMISSIONS

Peking University

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