  We investigate Aharonov-Bohm scattering in a theory in which charged bosonic
matter fields are coupled to topologically massive electrodynamics and
topologically massive gravity. We demonstrate that, at one-loop order, the
transmuted spins in this theory are related to the ones of ordinary
Chern-Simons gauge theory in the same way that the
Knizhnik-Polyakov-Zamolodchikov formula relates the Liouville-dressed conformal
weights of primary operators to the bare weights in two-dimensional conformal
field theories. We remark on the implications of this connection between
two-dimensional conformal field theories and three-dimensional gauge and
gravity theories for a topological membrane reformulation of strings. We also
discuss some features of the gravitational analog of the Aharonov-Bohm effect.
