Möbius Inversion and Duality for Summations of Stable Graphs

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 Möbius function of a poset, we introduce generalized ($\mathbb{Q}$-valued) zeta function and generalized ($\mathbb{Q}$-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.