  The goal of this paper is to raise the possibility that there exists a
meaningful theory of `motives' associated to certain hypergeometric integrals,
viewed as functions of their parameters. It goes beyond the classical theory of
motives, but should be compatible with it. Such a theory would explain a recent
and surprising conjecture arising in the context of scattering amplitudes for a
motivic Galois group action on Gauss' ${}_2F_1$ hypergeometric function, which
we prove in this paper by direct means. More generally, we consider Lauricella
hypergeometric functions and show on the one hand how the coefficients in their
Taylor expansions can be promoted, via the theory of motivic fundamental
groups, to motivic multiple polylogarithms. The latter are periods of ordinary
motives and admit an action of the usual motivic Galois group, which we call
the `local' action. On the other hand, we define lifts of the full Lauricella
functions as matrix coefficients in a Tannakian category of twisted cohomology,
which inherit an action of the corresponding Tannaka group. We call this the
`global' action. We prove that these two actions, local and global, are
compatible with each other, even though they are defined in completely
different ways. The main technical tool is to prove that metabelian quotients
of generalised Drinfeld associators on the punctured Riemann sphere are
hypergeometric functions. We also study single-valued versions of these
hypergeometric functions, which may be of independent interest.
