  We study differential forms on an algebraic compactification of a moduli
space of metric graphs. Canonical examples of such forms are obtained by
pulling back invariant differentials along a tropical Torelli map. The
invariant differential forms in question generate the stable real cohomology of
the general linear group, as shown by Borel. By integrating such invariant
forms over the space of metrics on a graph, we define canonical period
integrals associated to graphs, which we prove are always finite and take the
form of generalised Feynman integrals. Furthermore, canonical integrals can be
used to detect the non-vanishing of homology classes in the commutative graph
complex. This theory leads to insights about the structure of the cohomology of
the commutative graph complex, and new connections between graph complexes,
motivic Galois groups and quantum field theory.
