Gravitational anyons, chern-simons-witten gravity and the gravitational aharonov-bohm effect

Abstract

The anyonic behaviour of massive point particles in the (2 + 1)-dimensional theory of topologically massive gravity, as first suggested by Deser, is further investigated. At the linearised level, we recall that the exotic statistics of the particles may be understood in terms of the gravitational Aharonov-Bohm effect discussed by several authors in the 1960's. This approach suggests that massive spinning point particles in ordinary Einstein gravity may also exhibit exotic statistics, and that it is the Einstein term which takes on the role carried out by the Chern-Simons term in the standard U(1) anyon model. We investigate this further in the Chern-Simons-Witten formulation of 2 + 1 gravity, where the Einstein term is indeed a Chern-Simons term. In this context the exotic statistics may be understood in terms of a non-abelian Aharonov-Bohm effect, giving a description in terms of the fully non-linear theory of gravity, and making explicit the relationship between gravitational anyons and Chern-Simons anyons of (2 + 1)-dimensional U(1) gauge theory. The analogue of magnetic flux in this context is seen to be both torsion and Riemann curvature. We finally show the correspondence between the statistics phase obtained in this approach and that obtained by Deser and McCarthy in the linearised approximation of topologically massive gravity.