Inducing Liouville theory from topologically massive gravity

Abstract

The well-known relationship between three-dimensional Chern-Simons theories and two-dimensional Wess-Zumino-Witten models may be understood as a straightforward consequence of symmetry breaking at the boundary of a three-manifold. In the standard Faddeev-Popov gauge fixing of a Chern-Simons path integral, the gauge group fails to decouple at the boundary, inducing instead a WZW action. From this point of view, the topological nature of Chern-Simons theory is no longer crucial. In particular, it is shown that (dynamical) topologically massive gauge theories give rise to WZW actions, and that topologically massive gravity induces a two-dimensional gravitational (Liouville) action. The equivalence of the three-dimensional local Lorentz anomaly and the associated two-dimensional conformal anomaly is demonstrated, and some new light is shed on the relationship between Liouville theory and the SL(2, R ) WZW model.