Superconformal SU(1,1|n) mechanics

TL;DR

This paper systematically studies multi-particle superconformal mechanics based on the SU(1,1|n) group, constructing representations, analyzing dynamics via prepotentials, and exploring solutions related to root systems, extending previous SU(1,1|2) results.

Contribution

It introduces a representation of the superconformal algebra su(1,1|n) for multi-particle systems and analyzes the associated dynamics and solutions, generalizing the SU(1,1|2) case.

Findings

Dynamics governed by prepotentials V and F

Solutions linked to root systems only produce decoupled models

Extension with angular variables faces significant challenges

Abstract

Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1,1|2) in mechanics. Remarking that SU(1,1|2) is a particular member of a chain of supergroups SU(1,1|n) parametrized by an integer n, here we begin a systematic study of SU(1,1|n) multi-particle mechanics. A representation of the superconformal algebra su(1,1|n) is constructed on the phase space spanned by m copies of the (1,2n,2n-1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1,2n,2n-1) supermultiplet by angular variables in a way similar to the SU(1,1|2) case is problematic.