On gauge theories for non-semisimple groups

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 a 1-loop term with a 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 the 4-dimensional algebra E₂^c. The quantum part of its effective action is expressed in terms of a 1-loop effective action of SU(2) gauge theory. The E₂^c model can also be described as a special limit of the SU(2) xU(1) YM theory with a decoupled ghost-like U(1) field.