Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

Let $\mathrm{\Gamma }$ be a connected bridgeless metric graph, and fix a point $v$ of $\mathrm{\Gamma }$. We define combinatorial iterated integrals on $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ at $v$ and the tensor algebra on the first homology of $\mathrm{\Gamma }$, $\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.