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arXiv · 2401.11717

M\"obius Inversion and Duality for Summations of Stable Graphs

Zhiyuan Wang·Jian Zhou
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Abstract

Using the stratifications of Deligne-Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus $g$ with $n$ external edges. By modifying the usual definition of zeta function and M\"obius function of a poset, we introduce generalized ($\mathbb Q$-valued) zeta function and generalized ($\mathbb Q$-valued) M\"obius function of the poset of stable graphs. We use them to proved a generalized M\"obius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.

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Zhiyuan Wang, Jian Zhou. 2024-01-22. M\"obius Inversion and Duality for Summations of Stable Graphs. https://arxiv.org/abs/2401.11717

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