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Möbius Inversion and Duality for Summations of Stable Graphs

Zhiyuan Wang, Jian Zhou

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.

Möbius Inversion and Duality for Summations of Stable Graphs

Abstract

Using the stratifications of Deligne-Mumford moduli spaces 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 with external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized (-valued) zeta function and generalized (-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.
Paper Structure (20 sections, 13 theorems, 99 equations)

This paper contains 20 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Duality in Abstract QFT for Stable Graphs
  3. Stable graphs and stratification of $\overline{{\mathcal{M}}}_{g,n}$
  4. Abstract free energies and abstract $n$-point functions
  5. Dotted stable vertices and dotted stable graphs
  6. The duality map and duality theorem for stable graphs
  7. Generalized Möbius Inversion Formula for Stable Graphs
  8. Partially-ordered set, incidence algebra, and Möbius inversion formula
  9. A partial ordering on the set of stable graphs
  10. Generalized zeta function and generalized Möbius function
  11. Generalized Möbius inversion formula
  12. Realization of the Generalized Möbius Inversion Formula
  13. Assigning Feynman rules to stable graphs
  14. Computation of the function $\tilde{g}$
  15. Realization of the inversion formula
  16. ...and 5 more sections

Key Result

Lemma 2.1

Denote by $\Gamma\in {\mathcal{G}}_{g,n}^c$ a stable graph (in the usual sense) without names, and $S_\Gamma$ the set of stable graph $\Gamma'$ with names on external edges, such that $\Gamma$ can be obtained from $\Gamma'$ by forgetting all the names. Then we have:

Theorems & Definitions (43)

  • Definition 2.1: wz
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.1: wz2
  • Definition 2.2: wz2
  • Example 2.4
  • Theorem 2.1: wz2
  • Theorem 2.2: wz2
  • Theorem 3.1: ro
  • ...and 33 more