Supersymmetry and Euler Multiplets

TL;DR

This paper explores solutions to Kostant's equation over specific cosets, revealing sets of Euler triplets that resemble supermultiplets but lack true supersymmetry, and constructs corresponding light-cone Lagrangians.

Contribution

It provides explicit solutions to Kostant's equation for two cosets, introduces Euler triplets, and develops free light-cone Lagrangians for these cases, advancing understanding of massless supermultiplet structures.

Findings

Solutions describe massless states with arbitrary spins

Euler triplets contain equal numbers of bosons and fermions

Constructed free light-cone Lagrangians for the models

Abstract

Some massless supermultiplets appear as the trivial solution of Kostant's equation, a Dirac-like equation over special cosets. We study two examples; one over the coset SU(3)/SU(2) times U(1) contains the N=2 hypermultiplet in (3+1) dimensions with U(1) as helicity; the other over the coset F_4/SO(9) describes the N=1 supermultiplet in eleven dimensions, where SO(9) is the light-cone little group. We present the general solutions to Kostant's equation for both cases; they describe massless physical states of arbitrary spins which display the same relations as the fields in the supermultiplets. They come in sets of three representations called Euler triplets, but do not display supersymmetry although the number of bosons and fermions is the same when spin-statistics is satisfied. We build the free light-cone Lagrangian for both cases.