LOOP REPRESENTATIONS FOR 2+1 GRAVITY ON A TORUS

We study the loop representation of the quantum theory for 2+1-dimensi onal general relativity on a manifold M = T2 x R, where T2 is the toru s, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts, and study its kernel. This kern el is dense in the connection representation, and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the co nnection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this constru ction, but note that certain ambiguities remain; in particular, functi ons of loops cannot be uniquely associated with functions of connectio ns.