Liouville Dressed Weights and Renormalization of Spin in Topologically Massive Gravity

We examine the relations between observables in two- and three-dimensional quantum gravity by studying the coupling of topologically massive gravity to matter fields in non-trivial representations of the three-dimensional Lorentz group. We show that the gravitational renormalization of spin up to one-loop order in these theories reproduces the leading orders of the KPZ scaling relations for quantum Liouville theory. We demonstrate that the two-dimensional scaling dimensions can be computed from tree-level Aharonov-Bohm scattering amplitudes between the charged particles in the limit where the three-dimensional theory possesses local conformal invariance. We show how the three-dimensional description defines scale-dependent weights by computing the one-loop order anomalous magnetic moment of fermions in a background electromagnetic field due to the renormalization by topologically massive gravity. We also discuss some aspects concerning the different phases of three-dimensional quantum gravity and argue that the topological ones may be related to the branched polymer phase of two-dimensional quantum gravity.