ON GAUGE-THEORIES FOR NONSEMISIMPLE GROUPS

We consider analogs of Yang-Mills theories for non-semisimple real Lie algebras which admit invariant non-degenerate metrics. These 4-dimens ional theories have many similarities with corresponding WZW models in 2 dimensions and Chem-Simons theories in 3 dimensions. In particular, the quantum effective action contains only a 1-loop term with a diver gent part that can be eliminated by a field redefinition. The on-shell scattering amplitudes are thus finite (scale invariant). This is a co nsequence of the presence of a null direction in the field space metri c: one of the field components is a Lagrange multiplier which 'freezes out' quantum fluctuations of the 'conjugate' field. The non-positivit y of the metric implies that these theories are apparently non-unitary . However, the special structure of interaction terms (degenerate comp ared to non-compact YM theories) suggests that there may exist a unita ry 'truncation'. We discuss in detail the simplest theory based on the 4-dimensional algebra E(2)(c). The quantum part of its effective acti on is expressed in terms of a 1-loop effective action of SU(2) gauge t heory. The E(2)(c) model can also be described as a special limit of t he SU(2) x U(1) YM theory with a decoupled ghost-like U(1) field.