  We present globally supersymmetric models of gauged scale covariance in ten,
six, and four-dimensions. This is an application of a recent similar gauging in
three-dimensions for a massive self-dual vector multiplet. In ten-dimensions,
we couple a single vector multiplet to another vector multiplet, where the
latter gauges the scale covariance of the former. Due to scale covariance, the
system does not have a lagrangian formulation, but has only a set of field
equations, like Type IIB supergravity in ten-dimensions. As by-products, we
construct similar models in six-dimensions with N=(2,0) supersymmetry, and
four-dimensions with N=1 supersymmetry. We finally get a similar model with N=4
supersymmetry in four-dimensions with consistent interactions that have never
been known before. We expect a series of descendant theories in dimensions
lower than ten by dimensional reductions. This result also indicates that
similar mechanisms will work for other vector and scalar multiplets in
space-time lower than ten-dimensions.
