N=4 mechanics, WDVV equations and roots

TL;DR

This paper explores N=4 superconformal multi-particle quantum mechanics, extending WDVV solutions associated with root systems, and classifies models with zero and nonzero central charge, including new dihedral group-based models.

Contribution

It extends WDVV solutions for N=4 mechanics using root systems and introduces new models based on orthocentric simplices and dihedral groups.

Findings

Classified models based on Coxeter root systems with zero central charge.

Constructed new models using orthocentric simplices and dihedral groups.

Explicitly derived full prepotentials for these models.

Abstract

N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.