Many-particle mechanics with D(2,1;alpha) superconformal symmetry

TL;DR

This paper constructs new N=4 superconformal many-particle models in one dimension using D(2,1;alpha) symmetry, solving complex equations for various root systems and invariance conditions.

Contribution

It introduces a method to realize D(2,1;alpha) superalgebra on particle systems with novel solutions for multiple particles and root systems, extending previous models.

Findings

Models based on deformed A_n and BCD_n root systems for any alpha.

Translation-invariant models for any number of particles at alpha=-1/2.

Special solutions for four particles at arbitrary alpha.

Abstract

We overcome the barrier of constructing N=4 superconformal models in one space dimension for more than three particles. The D(2,1;alpha) superalgebra of our systems is realized on the coordinates and momenta of the particles, their superpartners and one complex pair of harmonic variables. The models are determined by two prepotentials, F and U, which must obey the WDVV and a Killing-type equation plus homogeneity conditions. We investigate permutation-symmetric solutions, with and without translation invariance. Models based on deformed A_n and BCD_n root systems are constructed for any value of alpha, and exceptional F_n-type and super root systems admit solutions as well. Translation-invariant mechanics occurs for any number of particles at alpha=-1/2 (osp(4|2) invariance as a degenerate limit) and for four particles at arbitrary alpha (three series).