QUANTUM ASPECTS OF 2+1 GRAVITY

We review and systematize recent attempts to canonically quantize general relativity in 2+1 dimensions, defined on space-times $\mathbb{R}\times {\mathrm{\Sigma }}^g$, where ${\mathrm{\Sigma }}^g$ is a compact Riemann surface of genus $g$. The emphasis is on quantizations of the classical connection formulation, which use Wilson loops as their basic observables, but also results from the ADM formulation are summarized. We evaluate the progress and discuss the possible quantum (in)equivalence of the various approaches.