SCHOEBEL, K. and VESELOV, A.P., 2015. Separation coordinates, moduli spaces and Stasheff polytopes. Communications in Mathematical Physics, 37 (3), pp.1255-1274.
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 M¯0,n+2(R) of stable curves of genus zero with n + 2 marked points. We use the combinatorics of Stasheff polytopes tessellating M¯0,n+2(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 M¯0,n+2(R)
The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-015-2332-x