Gauging of Lorentz Group WZW Model by its Null Subgroup

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.

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