By Steven Carlip
Publisher: Cambridge University Press
Print Publication Year: 1998
Online Publication Date:December 2009
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511564192.001
Interest in (2+1)-dimensional gravity – general relativity in two spatial dimensions plus time – dates back at least to 1963, when Staruszkiewicz first showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. Over the next 20 years occasional papers on classical and quantum mechanical aspects appeared, but until recently the subject remained largely a curiosity.
Two discoveries changed this. In 1984, Deser, Jackiw, and 't Hooft began a systematic investigation of the behavior of classical and quantum mechanical point sources in (2+1)-dimensional gravity, showing that such systems exhibit interesting behavior both as toy models for (3+1)-dimensional quantum gravity and as realistic models of cosmic strings. Interest in this work was heightened when Gott showed that spacetimes containing a pair of cosmic strings could admit closed timelike curves; (2+1)-dimensional gravity quickly became a testing ground for issues of causality violation. Then in 1988, Witten showed that (2+1)-dimensional general relativity could be rewritten as a Chern–Simons theory, permitting exact computations of topology-changing amplitudes. The Chern–Simons formulation had been recognized a few years earlier by Achúcarro and Townsend, but Witten's rediscovery came at a time that the quantum mechanical treatment of Chern–Simons theory was advancing rapidly, and connections were quickly made to topological field theories, three-manifold topology, quantum groups, and other areas under active investigation.
Together, the work on point particle scattering and the Chern–Simons formulation ignited an explosion of new research.