Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

TL;DR

This paper introduces combinatorial iterated integrals on graphs, generalizing classical concepts, and demonstrates their ability to recover base points and characterize hyperellipticity through a harmonic volume in a tropical setting.

Contribution

It defines a unipotent combinatorial iterated integral framework on graphs, linking it to harmonic volumes and tropical Jacobians, and provides a new characterization of hyperellipticity.

Findings

The pairing recovers the base point up to finite ambiguity.

The harmonic volume encodes combinatorial and geometric data of graphs.

A potential-theoretic criterion characterizes hyperelliptic graphs.

Abstract

Let $\Gamma$ be a connected bridgeless metric graph, and fix a point $v$ of $\Gamma$. We define combinatorial iterated integrals on $\Gamma$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of $\Gamma$ at $v$ and the tensor algebra on the first homology of $\Gamma$, $\int\colon \mathbf{Z}\pi_1(\Gamma,v) \times T\mathrm{H}_1(\Gamma,\mathbf{R}) \to \mathbf{R}$. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point $v$ up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperelliptiicity for graphs.