Combinatorial Iterated Integrals and the Harmonic Volume of 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 $\Gamma$, $\int :\mathbf{Z}{\pi }_1(\Gamma ,v)\times T{\mathrm{H}}_1(\Gamma ,\mathbf{R})\to \mathbf{R}$. 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.

v3: Completely rewritten with changed title. 20pp. Comments always welcome!

References in corpus (1)

• Torelli theorem for graphs and tropical curves