Separation Coordinates, Moduli Spaces and Stasheff Polytopes

Abstract

We show that the orthogonal separation coordinates on the sphere S n are naturally parametrised by the real version of the Deligne–Mumford–Knudsen moduli space \({\bar{M}_{0,n+2}({\mathbb{R}})}\) of stable curves of genus zero with n + 2 marked points. We use the combinatorics of Stasheff polytopes tessellating \({\bar{M}_{0,n+2}({\mathbb{R}})}\) to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates, as well as of Stäckel systems from the mosaic operad structure on \({\bar{M}_{0,n+2}({\mathbb{R}})}\).