N=4 superconformal n-particle mechanics via superspace

TL;DR

This paper explores the construction of N=4 superconformal multi-particle systems in one dimension using superspace, deriving integrability conditions, and providing explicit solutions for three and four particles, including models related to WDVV equations.

Contribution

It introduces a systematic superspace approach to N=4 superconformal mechanics, deriving integrability conditions and explicitly constructing three- and four-particle models.

Findings

Derived integrability conditions for superpotentials and prepotentials.

Presented explicit three-particle models with translation invariance.

Constructed four-particle models using known WDVV solutions.

Abstract

We revisit the (untwisted) superfield approach to one-dimensional multi-particle systems with N=4 superconformal invariance. The requirement of a standard (flat) bosonic kinetic energy implies the existence of inertial (super-)coordinates, which is nontrivial beyond three particles. We formulate the corresponding integrability conditions, whose solution directly yields the superpotential, the two prepotentials and the bosonic potential. The structure equations for the two prepotentials, including the WDVV equation, follow automatically. The general solution for translation-invariant three-particle models is presented and illustrated with examples. For the four-particle case, we take advantage of known WDVV solutions to construct a D_3 and a B_3 model, thus overcoming a previously-found barrier regarding the bosonic potential. The general solution and classification remain a challenge.