$\mathcal N=4$ Multi-Particle Mechanics, WDVV Equation and Roots

We review the relation of $\mathcal N=4$ superconformal multi-particle models on the real line to the WDVV equation and an associated linear equation for two prepotentials, $F$ and $U$. The superspace treatment gives another variant of the integrability problem, which we also reformulate as a search for closed flat Yang–Mills connections. Three- and four-particle solutions are presented. The covector ansatz turns the WDVV equation into an algebraic condition, for which we give a formulation in terms of partial isometries. Three ideas for classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and matroids. Various examples and counterexamples are displayed.