In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{ M}_{ 0,n}$ 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 $\mathcal{ M}_{ 0,n}$, 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.
Categories: math.AG