Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 023, 21 pages      arXiv:1011.2207
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

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

Olaf Lechtenfeld, Konrad Schwerdtfeger and Johannes Thürigen
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany

Received November 14, 2010, in final form February 24, 2011; Published online March 05, 2011

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.

Key words: superconformal mechanics; Calogero models; WDVV equation; deformed root systems.

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