Genus zero modular operad and absolute Galois group

Abstract

In this article, we develop the geometry of canonical stratifications of the spaces \(\overline{\mathcal{M}}_{0,n}\) and prepare ground for studying the action of the Galois group \(Gal ({\overline{\mathbb{Q}}} /\mathbb{Q})\) upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces \(\overline{\mathcal{M}}_{0,n}\), equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An important feature is that 0-divisors of the split quaternion algebra imply additional singular points, and lead to investigations concerning the geometry and mixed Hodge structures.