Six ways to quantize (2+1)-dimensional gravity

We do not yet know how to quantize gravity in 3+1 dimensions, but in lower dimensions we face the opposite problem: many of the approaches originally developed for (3+1)-dimensional gravity can be successfully implemented in 2+1 dimensions, but the resulting quantum theories are not all equivalent. In this talk, I discuss six such approaches --- reduced phase space ADM quantization, the Chern-Simons/connection representation, covariant canonical quantization, the loop representation, the Wheeler-DeWitt equation, and lattice methods --- for the simple case of a spacetime whose spatial topology is that of a torus. A comparison of the resulting quantum theories can provide some useful insights into the conceptual issues that underlie quantum gravity in any number of dimensions. (Talk given at the Fifth Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ontario, May 1993)