The Constraints of 2+1 Gravity

The study of the gravitational field in 2+1 spacetime dimensions (2 space,1 time) has blossomed in the last few years into a substantial industry, after important contributions by Leutwyler [1], Deser, Jackiw and 'tHooft [2] and Witten [3]. It provides a means of studying the conceptual problems it shares with the four dimensional theory. Some of these are the role of diffeomorphism invariance, time, topology, etc. One of the advantages over the four dimensional theory is that it is computationally much easier, although there are many different interpretations at both the classical and quantum levels. I shall talk about the constraints of this theory, with emphasis on two approaches, namely the second order and first order formalisms, and make comparison with the four dimensional theory wherever possible, and finally, discuss an operator algebra approach that has been developed in the last few years in collaboration with T.Regge [4].