LIOUVILLE DRESSED WEIGHTS AND RENORMALIZATION OF SPIN IN TOPOLOGICALLY MASSIVE GRAVITY

We examine the relations between observables in two-and three-dimensio nal quantum gravity by studying the coupling of topologically massive gravity to matter fields in non-trivial representations of the three-d imensional Lorentz group, We show that the gravitational renormalizati on of spin up to one-loop order in these theories reproduces the leadi ng orders of the KPZ scaling relations for quantum Liouville theory. W e demonstrate that the two-dimensional scaling dimensions can be compu ted from tree-level Aharonov-Bohm scattering amplitudes between the ch arged particles in the limit where the three-dimensional theory posses ses local conformal invariance, We show how the three-dimensional desc ription defines scale-dependent weights by computing the one-loop orde r anomalous magnetic moment of fermions in a background electromagneti c field due to the renormalization by topologically massive gravity. W e also discuss some aspects concerning the different phases of three-d imensional quantum gravity and argue that the topological ones may be related to the branched polymer phase of two-dimensional quantum gravi ty. (C) 1997 Elsevier Science B.V.