  We describe a new realization of both global and local supersymmetry acting
in spaces of commuting and anticommuting differential forms. Unlike the
standard supersymmetry, it has Lorentz scalar transformation parameters. It is
related but is not reducible to standard supersymmetry. Reformulation of the
Standard Model with the new supersymmetry, called scalar supersymmetry, can be
achieved with the particle content of the SM. BRST symmetry is extended to
include scalar supersymmetry multiplets. Linear realization of scalar
supersymmetry with free fields or with fields interacting with background
gravity is described. Gauge interactions require non-linear realizations of
scalar supersymmetry and, except for SU(2), non-linear gauge-fixing conditions.
Requiring scalar supersymmetry of interacting action with the simplest chiral
multiplet can reduce the dynamical content of the theory to that of the SM: one
complex scalar, gauge fields, and three generations of Weyl spinors. At low
energies scalar supersymmetry is explicitly broken by gauge interactions.
However, in the asymptotically free case it becomes exact in the ultraviolet
limit. Thus it has the two most desirable features of softly broken standard
supersymmetry built in. Implementation of exact scalar supersymmetry in an
interacting string action is given.
