Supersymmetry and Euler Multiplets

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.