  Kontsevich's 1997 formula for the deformation quantization of Poisson
brackets is a Feynman expansion involving volume integrals over moduli spaces
of marked disks. We develop a systematic theory of integration on these moduli
spaces via suitable algebras of polylogarithms, and use it to prove that
Kontsevich's integrals can be expressed as integer-linear combinations of
multiple zeta values. Our proof gives a concrete algorithm for calculating the
integrals, which we have used to produce the first software package for the
symbolic calculation of Kontsevich's formula.
