Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

Francis Brown (1) , Clément Dupont (2)

1 MI - Mathematical Institute [Oxford]
2 IMAG - Institut Montpelliérain Alexander Grothendieck

We prove a recent conjecture arising in the context of scattering amplitudes for a `motivic' Galois group action on Gauss' ${}_2F_1$ hypergeometric function. More generally, we show on the one hand how the coefficients in a Laurent expansion of a Lauricella hypergeometric function can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter admit a `local' action of the usual motivic Galois group. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit a `global' action of its Tannaka group. We prove that these two actions are compatible. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]; Physique mathématique [math-ph]; Physique des Hautes Energies - Phénoménologie [hep-ph]

Francis Brown, Clément Dupont. Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions. Nagoya Mathematical Journal, 2023, 249, pp.148-220. ⟨10.1017/nmj.2022.27⟩. ⟨hal-02503524⟩