Spontaneously broken supersymmetric systems of the nonlinear fields and gauge fields

We consider $N=1\mathrm{}$ supersymmetric systems of nonlinear fields and gauge fields in (3 + 1)-dimensional space-time. The nonlinear fields take values on Kählerian complex manifolds. In a Lagrangian formulation of the systems based on Grassmann manifolds, which is a class of Kähler manifolds, we show explicitly that both gauge symmetry and supersymmetry are spontaneously broken. A general argument, in terms of counting of degrees of freedom, further shows that spontaneous gauge-symmetry breakdown is also necessarily accompanied by supersymmetry breakdown in systems based on other classes of Kähler manifolds. The resulting particle spectra of the systems have remarkable massless sectors consisting of gauge fields and fermions only.