Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions
Francis Brown, Clément Dupont
TL;DR
Brown and Dupont develop a dual, compatible motivic framework for Lauricella hypergeometric functions: a global, Tannakian viewpoint using twisted cohomology and motivic fundamental groups, and a local viewpoint via Taylor expansions in the exponents that lift to motivic polylogarithms. They prove a sharp coaction compatibility between the global and local realizations, show that metabelian quotients of generalised associators encode Lauricella beta-quotients, and extend the formalism to single-valued periods, yielding double-copy relations. The work unifies several strands—periods of cohomology with coefficients, twisted pairings, and single-valued period maps—in a cohesive motivic setting, with explicit results for the Gauss {}_2F_1 case and connections to Feynman-parameter integrals. The results suggest an underlying Lauricella motive and provide a robust toolkit for studying motivic Galois actions on hypergeometric-type integrals and their single-valued counterparts, with potential applications to scattering amplitudes and beyond.
Abstract
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.