N=4 Mechanics, WDVV Equations and Polytopes

TL;DR

This paper explores the mathematical structure of N=4 superconformal quantum mechanics, linking solutions to Coxeter systems and polytopes, and examines constraints on prepotentials in multi-particle models.

Contribution

It introduces a new framework connecting WDVV equations, Coxeter systems, and polytopes in N=4 superconformal mechanics, with detailed examples and analysis of prepotential constraints.

Findings

Solutions are encoded by Coxeter systems and polytopes.

Standard ansatz for U fails for all finite Coxeter systems.

Three-particle models based on dihedral root systems are more flexible.

Abstract

N=4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A_n and B_3 examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.