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$\mathcal N=4$ Multi-Particle Mechanics, WDVV Equation and Roots
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: superconformal mechanics; Calogero models; WDVV equation; deformed root systems. 
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Olaf Lechtenfeld; Konrad Schwerdtfeger; Johannes Thürigen. $\mathcal N=4$ Multi-Particle Mechanics, WDVV Equation and Roots. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a22/

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