Graph integrals, Feynman periods, and single-valued multiple zeta values

Abstract

The Borel classes generating the stable cohomology of the general linear group can be represented by invariant differential forms. It is known that pulling these forms back along a tropical Torelli map yields canonical convergent integrals associated to graphs, which are closely connected to the cohomology of $\mathrm{GL}_n$ and of graph complexes. A natural question is what numbers these graph integrals are. We answer this for primitive canonical integrals by showing that they coincide with a family of complex position-space integrals arising in deformation quantisation. As a consequence, canonical integrals of graphs evaluate to single-valued multiple zeta values. We further deduce that every single-valued multiple zeta value occurs as a rational linear combination of Feynman periods of graphs with massless propagators. Finally, in the commutative graph complex, our result implies that the two associated cocycles agree.