Liouville Dressed Weights and Renormalization of Spin in Topologically Massive Gravity

TL;DR

This paper explores the connection between 2D and 3D quantum gravity, showing how topologically massive gravity influences matter fields and reproduces known scaling relations, with implications for understanding quantum gravity phases.

Contribution

It demonstrates the gravitational renormalization of spin reproduces KPZ scaling and connects 3D topologically massive gravity to 2D quantum Liouville theory through scattering amplitudes.

Findings

Reproduces KPZ scaling relations at one-loop order.

Computes scale-dependent weights via anomalous magnetic moments.

Links topological phases of 3D gravity to 2D quantum gravity phases.

Abstract

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.