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

TL;DR

This paper explores the connection between N=4 superconformal multi-particle models, the WDVV equation, and algebraic structures, providing new solutions and classification methods for these integrable systems.

Contribution

It introduces a covector ansatz transforming the WDVV equation into an algebraic condition and develops three classification approaches: ortho-polytopes, hypergraphs, and matroids.

Findings

Presented three- and four-particle solutions.

Reformulated the integrability problem as finding flat Yang-Mills connections.

Developed three classification methods for WDVV solutions.

Abstract

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.