Gauging of Lorentz Group WZW Model by its Null Subgroup

Abstract

We consider the standard vector gauging of Lorentz group $ SO(3,1) $ WZW model by its non-semisimple null Euclidean subgroup in two dimensions $ E(2) $. The resultant effective action of the theory is seen to describe a one dimensional bosonic field in the presence of external charge that we interpret it as a Liouville field. Gauging a boosted $ SO(3) $ subgroup, we find that in the limit of the large boost, the theory can be interpreted as an interacting Toda theory. We also take the generalized non-standard bilinear form for $SO(3,1) $ and gauge both $ SO(3) $ and $E(2)$ subgroups and discuss the resultant theories.