Quadratic conformal supersymmetry

The non-appearance of predicted superparticles in high-energy particle experiments has placed severe constraints on candidate models of supersymmetry; in particular on the masses of the superpartners of known particles. Drawing on our earlier work investigating quadratic deformations of Lie superalgebras we present in this paper our recent findings that for certain extensions of space-time supersymmetries, namely the conformal superalgebra, there are representations without any superpartners (see J. Phys. A: Math. Theor. 51 (2018) 145203). This possibility arises due to a remarkable coincidence of the allowable quadratic extensions of the algebra, where one demands that a generalisation of the PBW theorem holds, and the minimal polynomial identity satisfied by the even subalgebra.