Computing Euler characteristics using quantum field theory

TL;DR

This work develops a quantum-field-theoretic framework to compute the virtual Euler characteristics of $\mathrm{Out}(F_n)$ and related graph complexes by turning graph-counting problems into 0-dimensional QFT partition functions. By representing graphs as labeled structures, exploiting the exponential formula for connected components, and employing Gaussian integral representations, the authors derive generating functions $\mathbf{F}(\hbar)$ and $\mathbf{X}(\hbar)$ whose coefficients encode $\chi$-invariants; they connect these to Kontsevich's orbifold Euler characteristics and to Bernoulli-number refinements. They obtain explicit formulas, notably $\chi(\mathcal{GC}_2^{(n+1)})=0$ for even $n$ and $-\dfrac{B_{n+1}}{n(n+1)}$ for odd $n$, and they express $\chi(\mathrm{Out}(F_n))$ via the generating function $\mathbf{Y}(\hbar)$ tied to leaf-marked forests and renormalized QFT. A renormalized topological QFT is proposed that encodes counterterms in $\tfrac{x}{2}+\mathbf{Y}(-\hbar e^{-x})$, offering a principled way to study the asymptotics and sign structure of these Euler characteristics, in particular their negative growth and connection to Kontsevich's constructions.

Abstract

This paper explains how to use quantum field theory techniques to find formal power series that encode the virtual Euler characteristics of $\mathrm{Out}(F_n)$ and related graph complexes. Finding such power series was a necessary step in the asymptotic analysis of $χ(\mathrm{Out}(F_n))$ carried out in the authors' previous paper.

Figures (2)

• Figure 1: A graph $G$ with its representation as triple $(H,E,V)$ below. Half-edges are depicted as short black lines, the partition $V$ is indicated with dotted blue circles and the matching $E$ of half-edges, with dotted orange lines.
• Figure 2: A half-edge labeled graph $lG$.

Theorems & Definitions (14)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 4.1
• proof