Orbifold Euler Characteristics of

{\overline{\mathcal{M}}}_{g,n}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline{{{\mathcal {M}}}}}_{g,n}$$\end{document}

Abstract

We compute the orbifold Euler characteristics of

{\overline{\mathcal{M}}}_{g,n}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{\mathcal M}_{g,n}$$\end{document}

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})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi (\overline{{\mathcal {M}}}_{g,n})$$\end{document}

. 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})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi (\overline{{\mathcal {M}}}_{0,n})$$\end{document}

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})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ({\overline{{{\mathcal {M}}}}}_{g,n})$$\end{document}

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})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ({{\mathcal {M}}}_{g,n})$$\end{document}

.

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)

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