  We consider analogs of Yang-Mills theories for non-semisimple real Lie
algebras which admit invariant non-degenerate metrics. These 4-dimensional
theories have many similarities with corresponding WZW models in 2 dimensions
and Chern-Simons theories in 3 dimensions. In particular, the quantum effective
action contains only 1-loop term with the divergent part that can be eliminated
by a field redefinition. The on-shell scattering amplitudes are thus finite
(scale invariant). This is a consequence of the presence of a null direction in
the field space metric: one of the field components is a Lagrange multiplier
which `freezes out' quantum fluctuations of the `conjugate' field. The
non-positivity of the metric implies that these theories are apparently
non-unitary. However, the special structure of interaction terms (degenerate
compared to non-compact YM theories) suggests that there may exist a unitary
`truncation'. We discuss in detail the simplest theory based on 4-dimensional
algebra E^c_2. The quantum part of its effective action is expressed in terms
of 1-loop effective action of SU(2) gauge theory. The E^c_2 model can be also
described as a special limit of SU(2) x U(1) YM theory with decoupled
ghost-like U(1) field.
