Decomposition in Coxeter-chambers of the configuration space of $d$ marked points on the complex plane

Interest in Conformal Field Theories and Quantum Field Theory lead physicists to consider configuration spaces of marked points on the complex projective line, $Conf_{0,d}(\mathbb{P})$. In this paper, a real semi-algebraic stratification of $Conf_{0,d}(\mathbb{C})$, invariant under Coxeter-Weyl group is constructed, using the natural relation of this configuration space with the space $Dpol_d$ of complex monic degree $d>0$ polynomials in one variable with simple roots. This decomposition relies on subsets of $Dpol_d$ forming a good cover in the sense of Cech of $Dpol_d$ and such that each piece of the decomposition is a set of polynomials, indexed by a decorated graph reminiscent of Grothendieck's dessins d'enfant. This decomposition in Coxeter-Weyl chambers brings into light a very deep interaction between the real locus of the moduli space $\overline{\mathcal{M}}_{0,d}(\mathbb{R})$ and the complex one $\overline{\mathcal{M}}_{0,d}(\mathbb{C})$. Using this decomposition, the existence of geometric invariants of those configuration spaces has been shown. Many examples are provided. Applications of these results in braid theory are discussed, namely for the braid operad.