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

[摘要] We review the relation of 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.

[关键词] superconformal mechanics;Calogero models;WDVV equation;deformed root systems