  N=4 superconformal n-particle quantum mechanics on the real line is governed
by two prepotentials, U and F, which obey a system of partial nonlinear
differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde
(WDVV) equation for F. The solutions are encoded by the finite Coxeter systems
and certain deformations thereof, which can be encoded by particular polytopes.
We provide A_n and B_3 examples in some detail. Turning on the prepotential U
in a given F background is very constrained for more than three particles and
nonzero central charge. The standard ansatz for U is shown to fail for all
finite Coxeter systems. Three-particle models are more flexible and based on
the dihedral root systems.
